(written from a Production point of view)
The Big Bang Theory (2007-2019) was a long-running CBS sitcom about four nerdy CalTech scientists (the insecure Leonard Hofstadter, the severely empathetically challenged Sheldon Cooper – who fancies himself a kindred spirit to Spock and Data – , the sexist, yet also mama's boy Howard Wolowitz, and the feminine but lovelorn Raj Koothrappali) and the hot promiscuous girl Penny who lives across the hallway and whom Leonard carries a torch for.
Being passionate comic book collectors, the four scientists are also rabid science fiction fans and collectors of memorabilia, Star Trek and Star Wars in particular, resulting in a constant flow of Star Trek references throughout the entire run of the show, particularly in its middle seasons. Other franchises the four are fans of include, among others, Stan Lee's Marvel Comics universe, Ronald D. Moore's Battlestar Galactica, and the British Doctor Who and Red Dwarf franchises, also referenced on the show, albeit far less frequently than Star Wars and Star Trek. Of all the Star Trek live-action incarnations, Star Trek: The Original Series, Star Trek: The Next Generation, and their derivative Star Trek films are the most frequently referenced, with the others barely touched upon, if at all – specifically Star Trek: Enterprise in the latter case. This in itself is reflective of the status these two productions hold in the awareness of the general (non-fan) populace. Similarly, Star Trek II: The Wrath of Khan is the most referenced film, likewise reflecting its status.
A major recurrent secondary character on the show is the socially awkward and consistently strapped-for-cash Stuart Bloom, owner of the comic book store the four protagonists frequent, and who, in an instance of cross-fertilization, has the in-universe animal Stuart named after him. (AT: "Choose Your Pain")
Recurrent guest stars on the show include Brian George and Keith Carradine as Raj's and Penny's fathers, Pasha Lychnikoff as a cosmonaut on the International Space Station, Mark Harelik as the CalTech physics department chairman Gablehauser, and Jerry O'Connell as Sheldon's older brother George, Jr., with Stephen Hawking as himself. Star Trek actors Wil Wheaton, Leonard Nimoy, George Takei, LeVar Burton, Brent Spiner, and ultimately William Shatner all made appearances on the show, as fictionalized versions of themselves, in Wheaton's and Burton's cases recurrent ones. Cast member Kaley Cuoco ("Penny") also played Shatner's daughter in Priceline.com commercials, starting in 2013 while The Bang Theory was in its sixth season,  and Iris Bahr appeared in the original unaired pilot for the show. Non-performing alumni with formal Star Trek ties who made (cameo) appearances as fictionalized versions of themselves included D.C. Fontana, David Gerrold, Adam Nimoy, Neil deGrasse Tyson, Elon Musk, and Stan Lee, the latter four in speaking parts.
Both the The Big Bang Theory and Star Trek franchises are (in part) owned by CBS Corporation, as is the 2017 spin-off prequel series Young Sheldon. Unlike the CBS-era Star Trek television productions though, The Big Bang Theory and its spin-off are produced on behalf of CBS by Warner Bros., which is therefore the end-owner of that franchise, like NBC and Desilu/Paramount Pictures had been in the Original Series era. To this day NBC still receives a residual income stream from the Original Series – predominantly from the syndicated airings and streamings – as CBS will from the The Big Bang Theory franchise.
When The Big Bang Theory concluded its twelve-season run in 2019, it was officially reported by a proud CBS as the longest running multi-camera sitcom in US television history,   though it was not the longest running live-action comedy series. 
Generic Star Trek references
Miscellaneous Star Trek merchandise
Even when Star Trek was not directly referenced to in the individual episodes, a multitude of its merchandise from various manufacturers was featured throughout the entire series as either props (most notably uniforms and handheld props, thereby constituting something akin to product placement), background set dressings in the homes of the four scientists or as merchandise in Stuart Bloom's comic book store. This merchandise originated from the archives of CBS Consumer Products, where it is habitually kept. Merchandise that is identified and/or warrants further attention is addressed in their respective individual episode entries, listed below.
"Wild Stacks" video slot machine
In 2015 Australian gaming developer Aristocrat launched a three reel slot machine called "Big Bang Theory Slot Game". One of the payout symbols is a small pile of Star Trek: The Next Generation comic books. 
Leonard Nimoy homage
Upon the the passing of Leonard Nimoy in February 2015, The Big Bang Theory paid tribute to the Spock performer in the season 8 episode "The Colonization Application", aired on 11 April 2015, by closing out with a final end title card that read "The impact you had on our show and on our lives is everlasting."
Aside from being referenced to on numerous occasions in prior (and later) episodes of the show, both as performer as well as his Spock alter ego, Nimoy had already made his voice-over appearance on the show in the season 5 episode "The Transporter Malfunction" which had aired on 29 March 2012.
Original unaired pilot
Season 1 (2007-2008)
"The Cooper-Hofstadter Polarization" (Ep. 09)
A presentation of a scientific paper by Leonard at a science conference goes awry, when co-author Sheldon who had opposed the presentation, starts an altercation with Leonard during which he threatens Leonard with a "Vulcan Nerve Pinch".
"The Loobenfeld Decay" (Ep. 10)
Penny is reconsidered for a part in the musical Rent and prepares for it by rehearsing singing the songs for the part. She invites Leonard and Sheldon to her performance, but they, horrified by the appalling quality of her singing, go out of their way to make up excuses for not going.
"The Pancake Batter Anomaly" (Ep. 11)
Sheldon and Leonard play three-dimensional chess.
"The Bat Jar Conjecture" (Ep. 13)
The episode begins with Howard announcing new info on the then-upcoming Star Trek film, which leads into a discussion about alien private parts and Spock's conception. Later Sheldon is persuaded to join in the Physics Bowl by quoting Spock's dying words. Sheldon "designs" shirts for his own Physics Bowl team, which bear an "uncanny" resemblance to TOS-style uniforms, with his teammates clad in red shirts and being otherwise prohibited to speak up during the contest, whereas Sheldon wears a command-gold shirt and is the only one allowed to answer questions – which had been the reason why he was ousted from his friend's team to begin with. Ultimately in the final scene, to satisfy who is smarter between Leonard and Sheldon, Penny asks multiple real-world questions for which they are only able to give Star Trek-like answers, including Patrick Stewart or William Shatner being the Sexiest Man Alive and Tweety Bird thinking he saw a Romulan.
"The Nerdvana Annihilation" (Ep. 14)
The guys purchase a full-scale time machine prop replica from the film The Time Machine and they decide to play with it. They wonder what time date to travel to and Sheldon suggests to Leonard that he should travel to the future to obtain a cloaking device and references "The Enterprise Incident". Sheldon gives the stardate of this episode as 5027.3, which supposedly converts to January 10th, 2328.
When a grumpy Penny viciously chides the boys for their "childish" passions, Leonard decides to get rid of his memorabilia collection, sparking an immediate fight among the three others over ownership of Leonard's possessions. When the fight starts to escalate, Leonard threatens to open up a Geordi La Forge action figure that's still in its original packaging. During the altercation, Howard is wearing a shirt that resembles a Starfleet Academy uniform and wears an enlistee/chief engineer's pip on the collar of his undershirt as rank insignia.
Season 2 (2008-2009)
"The Bad Fish Paradigm" (Ep. 01)
"The Codpiece Topology" (Ep. 02)
Sheldon Cooper goes to a Renaissance fair dressed up like Spock pretending he's on a planet with a Renaissance theme due to the fair's complete disregard for historical accuracy.
"The Euclid Alternative" (Ep. 05)
When Penny drives Sheldon to work, he annoys her by continuously criticizing her driving style. When a thoroughly irritated Penny rebukes him, Sheldon seemingly drops his attitude by stating, "You have the conn" – even though he has no intention whatsoever to do anything of the sort.
"The Panty Pinata Polarization" (Ep. 07)
"The Lizard-Spock Expansion" (Ep. 08)
Sheldon expands the game of Rock-Paper-Scissors with additional options of "Lizard" and "Spock," the latter symbolized by the Vulcan live-long-and-prosper hand gesture. It was used initially to determine whether or not to watch Star Trek: Deep Space Nine. Later, Sheldon and Raj argue whether Star Trek: The Motion Picture or Star Trek V: The Final Frontier is the worst Trek movie, and whether Star Trek II: The Wrath of Khan or Star Trek IV: The Voyage Home is the best.
"The White Asparagus Triangulation" (Ep. 09)
Sheldon compares Dr. Stephanie Barnett as the Leonard McCoy to their social group's "landing party" (Leonard is Kirk, Sheldon is Spock, Wolowitz is Scotty, Koothrappali is a redshirt) and gives a somewhat off version of the Vulcan salute when he notices her observing the conversation. He later tries to relieve Leonard of duty with Starfleet general order 104, section A. Leonard argues that it won’t work because they aren’t on Star Trek.
"The Cushion Saturation" (Ep. 16)
Sheldon shoots Penny at the weekly paintball expedition before the game has even started in retaliation for Penny having accidentally shot his favorite cushion with a paintball gun earlier. When asked why he would do this, Sheldon paraphrases Khan, stating that "revenge is a dish best served cold".
"The Bath Item Gift Hypothesis" (Ep. 11)
Penny gives Sheldon a used napkin autographed by Leonard Nimoy as a Christmas present. She apologizes because Nimoy wiped his mouth with it, but a more than overjoyed Sheldon is even more grateful, as now he has Nimoy's DNA, claiming he now will soon be able to "grow" a Leonard Nimoy for himself.
"The Dead Hooker Juxtaposition" (Ep. 19)
Penny compares the main characters' social naivete to not knowing how to put up their shields like on Star Trek.
"The Hofstadter Isotope" (Ep. 20)
"The Vegas Renormalization" (Ep. 21)
Sheldon's frequent answer of Spock during Twenty Questions.
"The Classified Materials Turbulence" (Ep. 22)
Leonard says that Howard's zero-gravity toilet for the International Space Station will allow its crew "to boldly go where no man has gone before."
Season 3 (2009-2010)
"The Electric Can Opener Fluctuation" (Ep. 01)
Penny references Star Trek as part of her comforting of Sheldon after he learns Leonard, Howard, and Raj had faked his discovery of magnetic monopoles in order to keep him happy. Sheldon wears a goatee resembling the one Spock wears in TOS: "Mirror, Mirror". A recurring joke is Sheldon's "damned Vulcan hearing."
"The Creepy Candy Coating Corollary" (Ep. 05)
Leonard says that Howard can speak five languages, and Howard says, “Six, if you count Klingon.” Sheldon states he used to be a Wil Wheaton fan until Wil abruptly pulled out of a convention Sheldon had traveled ten hours by bus to attend and get his autograph. During a customizable card game tournament Wil is participating in, Sheldon constantly quotes Khan Noonien Singh from Star Trek II: The Wrath of Khan, uttering the proverb "Revenge is a dish that is best served cold" in original Klingon. When Wheaton tricks Sheldon out of a sure victory, Sheldon cries out "Wheaton" to the heavens, in imitation of Kirk's "Khan" cries in The Wrath of Khan.
"The Adhesive Duck Deficiency" (Ep. 08)
Trying to get Sheldon to drive faster, Penny says, "warp speed ahead, Mr. Spock", to which Sheldon replies that that order would never be given to the science officer, but to the navigator/helmsman instead.
"The Maternal Congruence (Ep. 11)
When Leonard asks what he truly wants, Sheldon answers, "What I want is to be departing on the starship Enterprise in a one-man shuttlecraft headed for the planetoid I rule, known as Sheldon Alpha Five."
"The Bozeman Reaction" (Ep. 13)
"The Large Hadron Collision" (Ep. 15)
In order to persuade Leonard, who is planning to take Penny along as a Valentine's Day gift, to take him instead to the CERN Large Hadron Collider in Switzerland, Sheldon goes even as far as allowing Leonard to watch Babylon 5. Prohibited to be seen by Sheldon, Leonard loves the series but had until then lost each and every argument they had over it. However, Sheldon is so desperate to go to CERN that he purports to set aside his disdain for the series, even though it, according to him, "(...)failed as drama, science fiction and is hopelessly derivative".
"The Excelsior Acquisition" (Ep. 16)
Sheldon reveals that Leonard Nimoy took a restraining order out against him after receiving one from Stan Lee. Additionally, when Leonard is asking Penny to guess who Stan is, she incorrectly assumes he was on Star Trek.
"The Precious Fragmentation" (Ep. 17)
"The Pants Alternative" (Ep. 18)
Assuming he was passed over for a science award, Sheldon compares himself to the William Shatner of science after musing why Shatner was not in Star Trek, the 2009 alternate reality film. When Sheldon learns that he is the recipient of the award after all, he immediately reverts to his self-absorbed self again, causing Howard to scoff, "Yeah, the one thing the William Shatner of theoretical physics needed was an ego boost." 
"The Wheaton Recurrence" (Ep. 19)
"The Staircase Implementation" (Ep. 22)
"The Lunar Excitation" (Ep. 23)
As Howard and Raj, having secretly signed Sheldon up with an online dating service, try to convince him to at least meet the woman he has been matched with, Raj reminds him that even Spock dated once every seven years, so Sheldon quickly shoots back that pon farr had nothing to do with dating: "His blood boiled with mating lust".
Season 4 (2010-2011)
Season 4 promotional ads
In one, Sheldon builds an Original Series transporter to beam them (with Next Generation-style visual effects) from their old Monday night time slot to Thursday. In another he and Leonard again play three-dimensional chess while discussing the move.
"The Hot Troll Deviation" (Ep. 04)
Howard has a USS Enterprise-D model in his bedroom. An imaginary George Takei (wearing his Hikaru Sulu costume from Star Trek IV: The Voyage Home) appears to give Howard romantic advice. Likewise, actress Katee Sackhoff appears as her Battlestar Galactica counterpart, for whom Howard has the hots, and starts bickering with Takei, questioning his credentials to give Howard romantic advice, while poking fun at Takei's homosexuality. Both however, are also lamenting and exchanging notes on being typecast as Sulu and Starbuck respectively.
"The Apology Insufficiency" (Ep. 07)
Taken as proof that he has wronged Howard, Sheldon is dreaming of a Gorn sitting on the couch reading a magazine. He later compares his problem of Howard refusing to accept his apology with the no-win Kobayashi Maru scenario, and is impressed by Penny's knowledge that Kirk cheated on the test, enticing him to try something similar on Howard. Later he is introduced by Raj to the visiting Neil deGrasse Tyson at CalTech, but Sheldon scolds him for his part in the declassification of Pluto from a planet to dwarf planet, and refuses to accept his apologies for it later on, stating that Tyson needs "to get over himself".
"The 21-Second Excitation" (Ep. 08)
Leonard mentions how they all waited fourteen hours to see Star Trek Nemesis, also relating how he had to fight a "Klingon" during the wait, the ordeal only aggravated by the fact that they had to watch the "terrible movie" afterwards. Later, when Wil Wheaton – deemed the Jar Jar Binks of the Star Trek universe by Sheldon – jumps the line to see Raiders of the Lost Ark, Sheldon quotes Picard's "The line must be drawn HERE!" line from Star Trek: First Contact.
"The Boyfriend Complexity" (Ep. 09)
Sheldon comments that he doesn't care for novelty editions of Monopoly, preferring the regular and Klingon editions.
Howard and Raj play a game of Star Trek-themed battleships. When Howard notices that two squares are a hit but the one in the middle is a miss, Raj comments that his starship has a hole in the middle. When asked what starship has a hole in the middle, Raj says "a Romulan battle bagel."
"The Alien Parasite Hypothesis" (Ep. 10)
"The Bus Pants Utilization" (Ep. 12)
Sheldon's passive-aggressive stance against the guys for kicking him out of their app design team includes playing the Original Series theme on a theremin. Before he was kicked out Sheldon listed himself as the "Star Trek science officer" of the design team.
"The Love Car Displacement" (Ep. 13)
Leonard uses the Romulan Neutral Zone as a metaphor for how he and Penny will share a bed without becoming intimate. When he suggests they do get intimate in the context of how often the Neutral Zone was entered by both the Federation and Romulan Star Empire, Penny says she's raised shields over the lower half of her body.
"The Thespian Catalyst" (Ep. 14)
"The Toast Derivation" (Ep. 17)
Sheldon invited LeVar Burton over to the apartment via Twitter. At the end of the episode, he sticks his head in and gets too weirded out by the sight of three other guys singing karaoke. He then leaves and says he's so done with Twitter.
"The Prestidigitation Approximation" (Ep. 18)
Leonard says to Priya that the only Star Trek uniforms he has are the everyday and dress ones from the Original Series, when she peruses trough his wardrobe closet, coming across them. She then reveals that Raj has an Uhura dress, much to the family's alarm and dismay, when Leonard suggests she wear one.
Among the many science fiction memorabilia Leonard keeps in his bedroom besides the uniforms, is a 6, or 7" display figurine of Captain Kirk on his night dresser, as well a smaller Star Trek bust on his cubbard.
"The Zarnecki Incursion" (Ep. 19)
Sheldon carries a bat'leth with him to the house of the guy who hacked his World of Warcraft account.
"The Agreement Dissection" (Ep. 21)
Sheldon cites "Let That Be Your Last Battlefield" as his inspiration for blackmailing Leonard into signing a new roommate agreement that favors him. He then shows an okudagram-styled auto-destruct-esque countdown timer on his laptop showing how much time Leonard has before Priya's parents are emailed about her dating Leonard. Sheldon then tells her that while she may have gone to the Faculty of Law at Cambridge University, he's an honorary graduate of Starfleet Academy.
"The Wildebeest Implementation" (Ep. 22)
Sheldon's three-player variation of chess includes a space designated as a transporter pad.
"The Roommate Transmogrification" (Ep. 24)
Raj overhears Leonard and his sister Priya having an Original Series-themed sex fantasy.
Season 5 (2011-2012)
"The Russian Rocket Reaction" (Ep. 05)
Wil Wheaton throws a party at his house, which has a painting of him as Wesley Crusher in uniform and an Enterprise-D model. Raj says he talked with Alice Krige's gynecologist, referring to Alice as "The Borg Queen From First Contact." Wil gives Sheldon an autographed still-in-package Playmates Toys Wesley action figure, but Brent Spiner promptly opens it upon seeing it. Brent then offered Sheldon a autographed still-in-package Data action figure to make up for it, but Sheldon declines and declares Spiner as his new mortal enemy. The other boys though, manage to negotiate Spiner down on the price of his action figure from twenty dollars to two for fifteen and a promise that Spiner will attend a birthday party.
"The Rhinitis Revelation" (Ep. 06)
There's a Next Generation science tricorder displayed in the bookcase behind the couch.
"The Good Guy Fluctuation" (Ep. 07)
Dispelling his claim that he is too intelligent to be surpised by Halloween pranks, Sheldon is literally scared senseless when Leonard does surprise him wearing a Balok mask.
"The Ornithophobia Diffusion" (Ep. 09)
While the two are on a date in a bar, they start bickering and Penny tells a girl that Leonard owns two Star Trek uniforms and that he wears them outside the Halloween holidays as well.
"The Recombination Hypothesis" (Ep. 13)
Sheldon ordered a life-size cardboard standee of Spock for his room, but the company sent him the wrong version, prompting him to snarl "Live long and suck it, Zachary Quinto!" – despite Leonard reminding him that he had liked Quinto as "Sylar" in Heroes.
"The Vacation Solution" (Ep. 16)
Bernadette had rejected Howard's suggestion to have wedding invitations in Klingonese. Sheldon later mentions in passing that Carl Sagan is yet another celebrity who has taken out a restraining order against him.
"The Rothman Disintegration" (Ep. 17)
Kripke has Sheldon explain Rock-Paper-Scissors-Lizard-Spock to him at least three times to mess with him. Sheldon later proposes a trivia contest with The Original Series, The Next Generation, Deep Space Nine, and Voyager being four of the five categories.
"The Weekend Vortex" (Ep. 19)
Amy mistakenly refers to the Star Wars video game the guys are playing as a Star Trek video game. Penny corrects her, saying the guys don't like when the two franchises are mistaken for one another, even though Penny, as far as she herself is concerned, does not perceive any difference between the two either.
"The Transporter Malfunction" (Ep. 20)
Penny bought Sheldon and Leonard a mint-in-box 1975 Star Trek transporter toy each. However, they refused to play with them as the toys would lose some of their perceived value within toy collector circles. That night, Sheldon dreamed that his Mr. Spock action figure told him that it was illogical to own a toy that he wouldn't play with. Shortly afterwards, Sheldon accidentally broke his transporter while playing with it and then replaced it with Leonard's. Later, he had a nightmare about being chastised by Mr. Spock about his actions and was subsequently attacked by a Gorn. Appropriately chastened, Sheldon confessed his guilt to his roommate.
"The Hawking Excitation" (Ep. 21)
Howard says "no" in Klingonese when telling Sheldon that he wouldn't introduce him to Stephen Hawking.
"The Stag Convergence" (Ep. 22)
Sheldon tells Wil Wheaton (who said he was attending Howard's bachelor party because it was a welcome alternative to George Takei's hot tub parties) to "talk to the hand" and gives him the Vulcan salute.
"The Launch Acceleration" (Ep. 23)
Howard tells Raj "You watch Star Trek, I live it" when discussing Howard's aborted mission to the International Space Station. Leonard and Sheldon again play three-dimensional chess. Later, Leonard comes home to find Sheldon and Amy playing "Star Trek doctor" with Sheldon uncomfortably, yet unwilling to stop, lying on the couch while Amy, dressed in a sciences division blue Original Series women's uniform, pretend-scans him with a toy medical tricorder.
"The Countdown Reflection" (Ep. 24)
Despite his initial offer to do so being soundly rejected, Sheldon performs part of Howard and Bernadette's wedding in Klingonese and says "and the Klingon High Council" when pronouncing them man and wife.
When the rocket is launched, Sheldon quotes the title sequence from The Original Series saying "Boldly go, Howard Wolowitz".
Season 6 (2012-2013)
"The Habitation Configuration" (Ep. 07)
Sheldon tries to do a segment on Trek flags on his "Fun With Flags" web show, first with Wil Wheaton (who keeps trying to do a bit on the Federation flag, much to Amy's dislike) and later with LeVar Burton, telling his viewers to "set phasers to fun." To get Amy to see things his way in the matter, Sheldon gives her the TNG Complete Series DVD Boxset and "briefly" describes "Encounter at Farpoint" scene-by-scene to her through her closed apartment door. Wil's house number is 1701, the Enterprise's hull number.
"The Fish Guts Displacement" (Ep. 10)
Amy suggests that Sheldon punish her by not letting her see the new Star trek movie, which Sheldon however finds a too harsh punishment not befitting the "crime" – to the dismay of Amy.
"The Egg Salad Equivalency" (Ep. 12)
When the two are discussing their day over dinner, Leonard mentions to Penny that he found an online quiz called "Which Star Trek character are you?", proudly stating that it only took him four tries to get Captain Kirk. When he starts to use Kirk's intonation, Penny cuts him short.
Later, when Leonard reassures an insecure and jealeous Penny that she has nothing to fear from Alex – the pretty research assistant of Sheldon who had made a move on Leonard earlier on – he does admit feeling like Captain Kirk, being chased by so many pretty women. Penny warns him that if he does not quit invoking Kirk right away, all womenhood will immediately stop doing so forthwith. "Message received", Leonard answers, not able to suppress the urge to use Kirk's intonation yet again.
"The Bakersfield Expedition" (Ep. 13)
While driving to the Bakersfield, California Comic-Con, the guys stop at Vasquez Rocks Natural Area Park to pose for photos in their The Next Generation costumes: Leonard as Picard, Sheldon as Data, Raj as Worf, and Howard as a Borg. After their car gets stolen, Sheldon states that they're all a landing party, obligated as such to overcome hardships. Failing to get a lift and being thrown refuge at by a passing car, its driver shouting "nerds" at them, the boys eventually stumble into a roadside restaurant in their disheveled Star Trek outfits, being met with wary bemusement by the other patrons, chuckling at the "Trekkies" – particularly after the "Borg" Howard asks if his mother should pick them up. The ordeal causes the despondent boys to question their love for Star Trek. After it is decided to rent a car instead to return home, Raj, more steadfastly holding on to his love, asks if the car can be from the rental agency Enterprise.
At the same time the boys have their ordeal, Chris Hemsworth's Marvel Comics character Thor becomes a point of argument for Penny, Amy, and Bernadette, when they made an effort to understand their boyfriends' nerdiness. When the boys return home, weary of being Trekkies because of their ordeal, they incredulously overhear their girlfriends' heated nerd discussion, causing their own inner nerds to be revitalized. On the assumption they have stumbled into an alternate reality and with their disheveled costumes put in order, the revived landing party prepares to break into the room to investigate with "phasers on stun", since Leonard in particular does not want to "vaporize" Penny, as he will never be able again to get another girlfriend that gorgeous.
"The Contractual Obligation Implementation" (Ep. 18)
While discussing the advancement of women in science, Sheldon names Star Trek writer D.C. Fontana as a woman who had to use her initials to hide her gender in her field of work. He also states he is preferential to a gender-blind society like in Star Trek, which is put in perspective by Leonard who observes that while warp drive was invented in that society, a black woman still had to answer the phone. Leonard's observation entices Howard to chime in with him reminiscing about how "many times he spent alone with Lt. Uhura in the shower."
"The Love Spell Potential" (Ep. 23)
Sheldon says he's tired of Leonard always invoking Zachary Quinto whenever Sheldon doesn't like an impending change.
"The Bon Voyage Reaction" (ep. 24)
Sheldon explains his personal trick to overcoming his social anxiety was to picture everyone as a Star Trek character. Apparently Raj is "Uhura" while Leonard is "Unnamed Crew Member In A Red Shirt".
Season 7 (2013-2014)
"The Hofstadter Insufficiency" (Ep. 01)
During Leonard's absence, Penny and Sheldon play a game of 3D chess. Sheldon also starts to ask Penny about Starfleet captains, before being cut off by her.
"The Discovery Dissipation" (Ep. 10)
To help Sheldon feel better about Leonard disproving his discovery of a new element, Wil Wheaton comes by and tells Sheldon that for a while he regretted being Wesley Crusher.
"The Cooper Extraction" (Ep. 11)
Sheldon is absent during the Christmas season, and while the gang is decorating the Christmas tree, Raj asks if the tree can now sport both Star Wars and Star Trek ornaments on the same branch, to which Leonard answers, "I know, it’s crazy. Welcome to the Thunderdome, people."
Season 8 (2014-2015)
"The Expedition Approximation" (Ep. 06)
During a deep cave simulation in an abandoned mine, Raj makes a remark about Voyager, Sheldon inquires which Voyager Raj means, the actual deep space probe or Star Trek: Voyager, the television show. After Raj confirms he is talking about the actual space probe, Sheldon retorts, "Good. Because I am too hot and tired to go on about how much I hate Voyager the TV show." Meanwhile, Howard and Bernadette have a heated argument on what each spouse brings into the marriage, with the high-earning Bernadette scoffingly accusing Howard to have only brought in his Star Trek collectible plate collection, and an incomplete one at that. Howard indignantly retorts he just completed the collection with the purchase of the last missing one on eBay, but tries to dodge Bernadette's inquiry for how much, after which the argument flares up again in full force.
"The Misinterpretation Agitation" (Ep. 07)
Sheldon offers a beverage to a Dr. Oliver Lorvis after he admits his intention to give some flowers to Penny. Later when receiving his drink, Lorvis, turning out to be a rabid science fiction memorabilia collector himself, compliments Sheldon's Star Trek mug and mentions that he got an original phaser, after having given Gene Roddenberry a vasectomy. Lorvis also divulges that he had helped William Shatner with kidney problems, having received from him an original tribble for his troubles, though he was not sure if it in effect had not been Shatner's toupee all along.
"The Champagne Reflection" (Ep. 10)
Sheldon introduces LeVar Burton for the final "Fun With Flags" webisode by saying he "puts the Geordi in your La Forge." Later LeVar says he should've taken Wil Wheaton's advice about getting a gate for his house when Sheldon tells him the web show's been resurrected.
"The Clean Room Infiltration" (Ep. 11)
At Amy's Christmas party, Dr. Koothrappali (Raj's father, who has recently divorced and is visiting from India) states that it is too soon to be dating again. Penny says she would if she was free, but pauses to inquire if he is not also talking about Star Trek all the time.
"The Colonization Application" (Ep. 17)
After he has wronged Amy, Sheldon tries to make amends by suggesting to play The Original Series theme through his nose. When he starts to do so, a still irked Amy cuts him short.
Season 9 (2015-2016)
"The Spock Resonance" (Ep. 07)
Wil Wheaton and Adam Nimoy come over to interview Sheldon for Adam's documentary about Leonard Nimoy, or as Sheldon put it, a "Spockumentary", quickly attributing the too obvious witticism to his roommate Leonard when it does not solicit any laughs. Sheldon tells the pair that the episode that really had meaning in his life was "The Galileo Seven". He also mentions that as a child he tried to perform a Vulcan neck pinch on a bullying schoolmate, which however only netted him a broken collarbone. He also divulges that he has left Wheaton the Wesley Crusher action figure in his will, the one he was gifted by him earlier in "The Russian Rocket Reaction". During their visit, Sheldon presents the napkin complete with Nimoy's DNA that he received from Penny as gift in "The Bath Item Gift Hypothesis". While interviewed, Sheldon's recent breakup with Amy becomes topic of the conversation, which he at first claims does not emotionally affect him, as he, like Spock, is above human emotions. When Penny – of all people – however points out to Sheldon that Spock was half-Human, and thus in part an emotional creature, Sheldon, clearly struggling with his feelings for Amy and unable to dispute Penny's "logic", breaks down and storms out of the room.
Later that evening, Sheldon reemerges from his room and informs Leonard and Penny that he is going to propose to Amy (which he until then unbeknownst to the others had already intended before the breakup) and get it over with either way, citing the Vulcan proverb "Kup-fun-tor ha'kiv na'ish du stau?" ("Can you return life to what you kill?"). Penny asks Leonard, if he understands what Sheldon is saying which he firmly denies, though timidly restating "no" in Vulcan ("nirsh") when she gives him the stink-eye. Before leaving, Sheldon exclaims that Amy can just "ponfo mirann" ("go to hell") if she should refuse him. "He doesn't mean that," Leonard quickly says to Penny when she again looks at him quizzically, completely destroying his credibility. Sheldon leaves for Amy's apartment, but finds her kissing another man in front of her building. After returning, a despondent Sheldon re-watches "Where No Man Has Gone Before", scoffing at Spock for his claims of being unemotional at the beginning of the episode.
"The Mystery Date Observation" (Ep. 08)
In an effort to get over Amy, Sheldon solicits the help of Howard and Raj to help him find a new mate, as they had already managed to do so with Amy through a dating site. They decide on a slightly different approach this time in singling out women, who meet all of Sheldon's many, many specifications by devising an online contest in which women can win the opportunity to meet Sheldon in person after having solved a series of puzzles, devised by Sheldon. One of the puzzles is to translate a Klingon text in ancient Sanskrit. As they had already been earlier with Amy, Howard, and Raj are flabbergasted when a handsome female scientist actually shows up at Sheldon's door quoting the text in impeccable Klingon and in its Sanskrit translation. Furthermore, she seems to tick of all of Sheldon's other boxes, while chatting with him. However, much to the consternation of his friends, Sheldon turns her away as she showed up one minute after the set deadline of the test had expired, as well as having solved the test while being bored by a symposium on atomic spectroscopy, the latter being a deal breaker for Sheldon.
"The Platonic Permutation" (Ep. 09)
Leonard has a pair of Mr. Spock oven mitts, which he shows to Penny.
"The Opening Night Excitation" (Ep. 11)
"The Meemaw Materialization" (Ep. 14)
While discussing his recent reconciliation with Amy, Leonard "points" out to Sheldon that he owns two pairs of Spock ears, one for ComicCon and one for around the house.
"The Celebration Experimentation" (Ep.17)
Wil Wheaton gives a speech during Sheldon's birthday party, stating that while Sheldon may be a Star Trek fan, he has become a Sheldon fan and concludes with the traditional Vulcan "Live long and prosper" valediction. Sheldon turns to Howard, who had just given a more traditional birthday speech, saying, "That's how you do it, Wolowitz. Now you see why he's famous and you're not". A short time later Stephen Hawking also chimes in via Skype, awkwardly singing "Happy Birthday".
Season 10 (2016-2017)
"The Military Miniaturization" (Ep. 02)
During an attempt to sell a by them invented guidance system to the military, under the proviso not to use it for lethal purposes, Colonel Williams states, "Look guys, its just a guidance system. It's not like you're handing me the Death Star from Star Trek." Forced by his his friends to remain silent as not to endanger their pitch, Sheldon nearly suffers from a stroke, because of the inaccuracy. When Williams subsequently assumes Howard to be the brains behind the invention, praising his engineering skills and his education at MIT, Sheldon can no longer contain himself and blurts out, "I'm Dr. Sheldon Cooper. I'm the actual brains behind this project. Also, engineers aren't real scientists, MIT is a trade school, and the Death Star is from Star Wars not Star Trek. Otherwise, thank you for your service."
"The Cohabitation Experimentation" (Ep. 04)
When Sheldon exhibits trepidation to switch apartments in order for both boys being able to live together with their respective paramours, Leonard entices Sheldon to do so, by comparing the five week experiment to The Original Series, stating, "The Enterprise was on a five-year mission to explore new worlds. Think of this as your own personal five-week mission to do the same," to which Shelon replies, "If you want to lure me in with sexy talk, that's how you do it." Very pleased with his own Star Trek wit, a smug Leonard is quickly put back in his place when Penny says that it is nothing to be proud of.
"The Veracity Elasticity" (Ep. 07)
Leonard discovers that Penny has been surreptitiously moving his collectibles into storage for quite some time now, one collectible at the time. One of the now missing items concerns his Klingon phrase-of-the day calendar. Not wanting to let Penny and Amy (who has also been lying to Sheldon that her apartment has long since been repaired as she does not want to end the "cohabitation experimentation" they embarked upon in episode 04) know that they are on to them, Leonard and Sheldon start to discuss the matter in Klingonese, much to the frustration of the girls, who subsequently start discussing themselves in Amy's own made-up language. When the Wolowitz couple arrives, Howard sides with the other boys and starts conversing in Klingon as well, before the girls, in unison, make them shut up.
"The Geology Elevation" (Ep. 09)
Sheldon is jealous of his CalTech geology colleague Bert, who has won an important science award. In a fit of frustration Sheldon tries to give him what he called a "Captain Kirk karate chop", but only succeeds in hurting his own hand. Back in the apartment Amy is bandaging his hand, when Stephen Hawking chimes in via Skype trying to console Sheldon. He reminds him that he has never won a Nobel Prize, but that he, like Sheldon, has many other life accomplishments and mentions his guest appearances on The Simpsons and Star Trek: The Next Generation. When an excited Leonard starts to explain his "Descent" scene to Penny, it takes her but one foul look to make him shut up.
"The Property Division Collision" (Ep. 10)
Now that the new living arrangements have become permanent, the time has come to divide the collectibles Sheldon and Leonard have accrued together in their time as roommates. Items to be divided include a Spock cuckoo clock and the 3D chess game which Sheldon can not possibly conceive Leonard wanting to have, as he has beaten him each and every time "in length, width and in depth". When Sheldon selfishly starts to lay claim on each and every single collectible, the boys start bickering. Not yet realizing that they are suffering from a severe case of separation anxiety after their thirteen years together, the conflicts quickly spirals out of control.
"The Holiday Summation" (Ep. 12)
During the holidays, Leonard has found another use for his pair of Spock ears besides ComicCon, wearing them as part of his Christmas elf outfit, when he and Penny go out to get a Christmas tree.
"The Allowance Evaporation" (Ep. 16)
While on a dinner dated with Amy, the subject of Sheldon's restraining orders comes up. Apart from the ones he already has from Carl Sagan, Leonard Nimoy, Stan Lee, and Bill Nye, it is revealed that Zachary Quinto too has taken one out against Sheldon, after he had tried to engage Quinto in conversation through the closed door of a toilet booth in a public restroom.
"The Long Distance Dissonance" (Ep. 24)
Sheldon is showing Dr. Ramona Nowitzki, a former admirer of Sheldon, his collection of letters he has received from celebrities. One of them is from Patrick Stewart in which he says that if Sheldon ever comes calling again at his house, he gets to "meet his dogs".
Season 11 (2017-2018)
"The Proposal Proposal" (Ep. 01)
After he has successfully proposed to Amy, Penny is musing if Sheldon is going to want "some weird Star Trek wedding", having already managed to successfully shut down hard Leonard's suggestion of having a "Doctor Who wedding cake".
"The Retraction Reaction" (Ep. 02)
"The Relaxation Integration" (Ep. 03)
Sheldon tells Amy that he will appear at their wedding with a tuxedo over a Star Trek uniform.
"The Explosion Implosion" (Ep. 04)
Leonard owns a Captain Kirk shirt, which needs to be dry cleaned.
"The Proton Regeneration" (Ep. 06)
Sheldon, who plans to audition for a role in a remake of "Professor Proton", visits Wil Wheaton in search for an acting coach, and asks him if he has the address of Patrick Stewart. Like Stewart has done before him, Wheaton threatens to set his dog on Sheldon, if he does not leave his premises immediately. Unbeknownst to Sheldon however, Wheaton is also vying for the role, and is put back on Sheldon's list of enemies after he becomes the one who gets the part.
"The Bitcoin Entanglement" (Ep. 09)
Having stewed on his retribution for years after the three other guys had mined bitcoins without him long before they gained their spectacular rise in value, Sheldon states, "I plotted my revenge. If you get a dish, I'll serve you some cold," when he comes clean that he had hid the files on Leonard's own flashdrive in retaliation, after the three desperate guys can not relocate the bitcoin files. Leonard then embarrasses Sheldon by disclosing that he had lost the flashdrive several years earlier.
"The Confidence Erosion" (Ep. 10)
When Sheldon and Amy start bickering over the details of their upcoming wedding ceremony, Sheldon insists on having their wedding vows uttered in Klingon if Amy does not dial back her own Little House on the Prairie-themed demands for the ceremony.
"The Sibling Realignment" (Ep. 23)
After having traveled to Texas, Sheldon and Leonard try to convince Sheldon's older brother George Jr. to attend his upcoming wedding, which the latter had refused at first. Later, George confides in private to Leonard that he had been hard on Sheldon in their childhood in order to protect him from his own foibles and to prepare him for real life. When Leonard asks if sitting on his younger brother's head while he was watching Star Trek had been part his tough love, George denies this by asserting that that had been "hilarious".
"The Bow Tie Asymmetry" (Ep. 24)
Wil Wheaton is slated to officiate the wedding of Sheldon and Amy, but his place is unceremoniously usurped by the by Howard arranged surprise guest Mark Hamil, the Luke Skywalker actor from Star Wars. Miffed, Wheaton acidly states, "So we finally meet", and when Hamil does not recognize Wheaton, the latter starts bickering with Hamil throughout the entire ceremony.
Season 12 (2018-2019)
"The Grant Allocation Derivation" (Ep. 07)
Sheldon soundly defeats Leonard yet again at a game of 3D chess, the latter's usual ineptitude at the game only aggravated by his preoccupation with an unenviable chore the university has burdened him with.
"The Confirmation Polarization" (Ep. 13)
The guys are discussing Star Trek: Discovery, Walking Dead, and Black Mirror during their lunch break in the university's cafeteria, but cut their discussion short because someone is behind in watching each and want to avoid hearing any spoilers.
"The D & D Vortex" (Ep. 16)
Will Wheaton brings William Shatner on his Professor Proton show to meet Sheldon, who unfortunately gets so excited to meet "Captain Kirk" as "Science Officer Cooper" that he vomits all over him. Later, the boys find out that Wheaton has Stuart, alongside Shatner and other celebrities, included as part of a Dungeons & Dragons role playing game group. Leonard, but not the other guys and certainly not Sheldon, gets invited to join the group after Stuart drops out. Though sworn to secrecy, Leonard cannot refrain himself from telling Penny – and thus the other two girls – who then accost Wheaton as one of the other celebrities is an actor the girls have the hots for. Leonard is subsequently thrown out by a miffed Wheaton, though he surreptitiously allows the girls to participate, as a form of punishment. During the game Shatner annoys his fellow gamers by making a constant flow of Star Trek references such as "I was about to go all Wrath of Khan on the ogres", and every time he does so, he has to put money in a Star Trek "swear jar", which he finds totally "worth it".
"The Decision Reverberation" (Ep. 20)
Spurred on by Penny to exert some much needed assertiveness, Leonard "confidently" tells Penny they are going to watch Discovery after they have had sex. Penny concedes but notifies Leonard she will be mostly watching her smartphone after sex.
Young Sheldon (2017-)
The Big Bang Theory universe was expanded significantly when in 2017, the prequel spinoff Young Sheldon was launched, focusing on the childhood of Sheldon Cooper in his home state of Texas, elaborating on the many childhood events Sheldon has mentioned over the course of the main series. Performer Jim Parsons reprised his role for the series as the adult Sheldon, retrospectively commenting on his childhood experiences in a voice-over part. The (recurrent) cast included Wallace Shawn as Dr. John Sturgis, the boyfriend of Cooper's maternal grandmother, affectionately coined "Meemaw" by Sheldon; Jason Alexander recurs as Gene Lundy, the high school drama teacher; while Harry Groener guest-starred in the first season episode "An Eagle Feather, A String Bean & An Eskimo" in 2018. Ed Begley, Jr. followed suit from the second season onward, recurring as Dr. Linkletter, colleague and rival of John Sturgis. The spin-off series starts off being set in 1989, two years after the launch of Star Trek: The Next Generation.
One of young Sheldon's favorite television programs he regularly watches on the show is Carl Sagan's acclaimed 1980 popular science series Cosmos: A Personal Voyage (as established in the main series Sagan will take out a retraining order against Sheldon later in life). That series was served by visual effects company Magicam, Inc., the company that had previously provided the studio models for Star Trek: The Motion Picture prior to the Cosmos assignment. Their work on Sagan's show had earned the company three Emmy Awards. An updated and modernized remake – or sequel – Cosmos: A Spacetime Odyssey, was aired in 2014, followed by Cosmos: Possible Worlds in 2020. Both produced by former Star Trek showrunner Brannon Braga, they were hosted by Neil deGrasse Tyson, who was inspired to embark on his scientific career because of Sagan and his show.
Whether or not (consciously) intended as such, young Sheldon's family background is actually remarkably similar to that of Star Trek creator Gene Roddenberry; like young Roddenberry, Sheldon is an atheist growing up in conservative Texas, has one brother and one sister, a devout Baptist mother who forces her offspring to attend religious gatherings, a non-religious father who is a Vietnam War veteran (Roddenberry's father was a World War I veteran), as well as having an aversion to his family situation. And like Roddenberry, Sheldon will go on to make a living for himself in California.
Season 1 (2017-2018)
"A Patch, a Modem, and a Zantac" (Ep. 06)
NASA engineer Dr. Ronald Hodges appears as a guest speaker in Sheldon's science class and does not take his idea about VTVL technology seriously. When cornered by Sheldon, Hodges tries to divert attention of his classmates by changing the subject to the sanitation technology for astronauts by asking the class, "Who wants to know where Captain Kirk has gone where no one has gone before?" Later Sheldon proves to Hodges that his theories are sound, but a chastised Hodges admits that technology has not advanced enough to implement his theories, after which a validated Sheldon lets the matter rest.
Decades later, it turned out that Elon Musk has surreptitiously used Sheldon's scientific notes, he somehow had acquired, to successfully have a booster rocket return to Earth on its own accord.
"Spock, Kirk, and Testicular Hernia" (Ep. 09)
Sheldon and Meemaw finish watching "The Return of the Archons". Later Sheldon uses Kirk's cheating on the Kobayashi Maru scenario to justify his own rule-breaking after Georgie admitted to cheating on his math test.
Season 2 (2018-2019)
"A Rival Prodigy and Sir Isaac Neutron" (Ep. 02)
Sheldon is joined in his class by another child prodigy, Paige, which does not sit well with him and he starts to experience bouts of jealously. His mother Mary on the other hand is overjoyed with the advent of another family with a gifted child and invites Paige and her parents to come over, hoping to exchange notes. Sheldon opposes this however, but Mary wins him over by pointing out to him that he had at first also opposed the new Star Trek series The Next Generation for not having Dr. Spock in it. After correcting his mother that it was Mr. Spock, he sees the "logic" of her argument but prepares himself for the visit by re-watching "Plato's Stepchildren" in which Dr. McCoy stated, "The release of emotions, Mr. Spock, is what keeps us healthy - emotionally healthy, that is," and intents to adhere to the Vulcan principle of kohlinar, just before his brother George, Jr. decides not to want to watch that "crap" and switches channel. Sheldon's intent comes to naught however, when Paige embraces him upon arrival, concluding that McCoy was idiotic in his observations. After Paige additionally and soundly beats him at a game of chess, Sheldon hears red alert bells go off in his mind's eye.
"A Crisis of Faith and Octopus Aliens" (Ep.3)
A local tragedy causes Mary to significantly increase her religious activities at first, with a retrospective Sheldon deeming his mother's revved-up religious fervor as akin to "starship Enterprise approaching warp 9".
"A Stunted Childhood and a Can of Fancy Mixed Nuts" (Ep. 10)
Missy taunts Sheldon with her statement that Paige, whom she has invited for a sleepover, speaks three languages in contrast to him speaking only his native one, after which he corrects his twin sister that he is in the process of learning conversational Klingon, continuing the conversation in it, before being cut short by Meemaw.
A model of the USS Enterprise NCC-1701 is suspended from the ceiling of the twins' bedroom.
Season 3 (2019-2020)
Season 3 featured updated opening titles footage; young Sheldon appears in different attires on an alternating basis, each reflecting one of his interests. One of these attires showed Sheldon dressed as Spock, complete with pointed ears, featured when an episode was heavy on Star Trek references, such as episode 11.
"An Entrepreneurialist and a Swat on the Bottom" (Ep. 3)
After Sheldon has misbehaved, angering his parents and Meemaw, he tries to awkwardly convey his feelings of remorse to his family by showing them a VHS recording of "The Devil in the Dark", using the miner's concession in that episode that they have wrongly mistreated the Horta as a proxy for his apology. His family does not get the reference.
"The Sin of Greed & A Chimichanga at Chi-Chi's" (Ep. 8)
Sheldon is offered full-time enrollment at the university where Sturgis teaches. Sheldon is seen watching a VHS recording of "The Enemy" where he mentions to George, Sr. how Geordi La Forge is lost on a planet and Wesley Crusher is using neutrinos in a probe to locate the missing chief engineering officer.
"A Live Chicken, A Fried Chicken & Holy Matrimony" (Ep. 11)
Much to Sheldon's chagrin, Georgie has taped over his VHS recording of "Final Mission" and replacing it with a Baywatch episode. To solve this dilemma, Sheldon uses the primitive internet technology of the day using message boards in the hopes of seeking out another Trekkie who may have an uncompromised copy.
After he has found a fellow Trekkie owning one, they both watch the episode at Meemaw's home, much to her chagrin. When they are done, Meemaw throws the Trekkie out, who scurries away, but who is "beaming out" in Sheldon's mind's eye.
"Pasadena" (Ep. 16)
Sheldon receives word that Stephen Hawking will be hosting a lecture at Caltech and finds himself begging both of his parents to take him to California and early on, they refuse, and in his moment of funk, Sheldon channel surfs only to find reminders of Hawking when seeing shows depicting characters in wheelchairs, chief among them, a scene of Christopher Pike in his wheelchair in "The Menagerie, Part I". By the second half, Sheldon is relieved when he and George Sr. board a flight so the child prodigy can hear his idol speak.