(covers information from several alternate timelines)
The crime of conspiracy was the act of collaborating with other persons to advance another criminal activity or other plot concealed from the general public. A cover-up was a type of conspiracy. Those who participated in a conspiracy were known as conspirators and the act of creating a conspiracy was known as conspiring.
In 2267, paranoid Starfleet officer Benjamin Finney, shortly after unsuccessfully trying to frame James T. Kirk, accused both Kirk and Starfleet in general of having conspired against him to prevent him being given command of the USS Enterprise. (TOS: "Court Martial")
In 2286, the crew of the USS Enterprise, led by Kirk, was charged with conspiracy after they retrieved Captain Spock from the Genesis Planet. The conspiracy, as charged, had led to acts of assault, sabotage, theft, and destruction of Federation property. All of the charges except for one were summarily dismissed by the President of the Federation. (Star Trek IV: The Voyage Home)
In 2364, a species of parasitic beings infiltrated Starfleet Command. Admiral Gregory Quinn was the first to suspect a conspiracy was going on at the highest levels of Starfleet. The alien plot was foiled by the efforts of the crew of the USS Enterprise-D and Admiral Norah Satie, among others. (TNG: "Coming of Age", "Conspiracy", "The Drumhead")
In 2367, suspected sabotage of a dilithium chamber hatch led to suspicion of a conspiracy onboard the Enterprise-D. Though it was ultimately determined to be an accident, Admiral Norah Satie began a wide-ranging inquiry into the potential conspiracy aboard the vessel, one Captain Jean-Luc Picard likened to a drumhead trial. The inquiry was brought to an abrupt halt when Satie went into a rage after Picard used own father's words against her and Vice Admiral Thomas Henry, disgusted, walked out of the proceedings. (TNG: "The Drumhead")
In 2368, Captain Picard became convinced that the crew of the Enterprise-D had become involved in a conspiracy after Ensign Ro Laren revealed to him that she had been assigned to the vessel on a secret mission by Admiral Kennelly under the apparently false pretense that Bajoran terrorists had attacked the Federation colony on Solarion IV. He decided to launch a secret mission of his own to reveal the truth. (TNG: "Ensign Ro")
That same year, Cadet Nicholas Locarno convinced fellow Nova Squadron members Sito Jaxa, Jean Hajar, and Wesley Crusher to participate in a (ultimately unsuccessful) cover-up of their attempt at the Kolvoord Starburst (which had resulted in the death of fellow member Joshua Albert) by pretending they had been executing a Yeager loop instead. (TNG: "The First Duty", "Lower Decks")
In 2369, the senior staff of Deep Space 9 talked about a way to arrest the Bajoran Kohn-Ma terrorist Tahna Los. Doctor Julian Bashir proposed that they could charge him with conspiracy, and Constable Odo mentioned that conspiracy was a crime. (DS9: "Past Prologue")
The same year, Commander Benjamin Sisko was able to prevent an assassination attempt on Vedek Bareil Antos. When he talked with Major Kira Nerys after this incident and the assassin Neela was interrogated, both Sisko and Kira believed they couldn't prove any connection to Vedek Winn Adami, and Neela was the only one who was accused of conspiracy. (DS9: "In the Hands of the Prophets")
When Seska stole mushrooms from Neelix's galley to prepare mushroom soup for Chakotay, the first officer learned about this criminal act while enjoying the soup. He thanked Seska for including him in this criminal conspiracy and reported it. (VOY: "State of Flux")
In 2375, The Doctor believed that Captain Kathryn Janeway's deleting his memories of Ensign Ahni Jetal and an attack by an unknown alien was part of a conspiracy of a conspiracy by her and the USS Voyager senior staff. In fact, Janeway was acting based on a decision made due to a conflict between The Doctor's cognitive and ethical subroutines, a decision The Doctor had agreed to, though his discovery ultimately forced her to reconsider her decision. (VOY: "Latent Image")
In 2377, Kadan conspired with the Quarren Ministry of Health and officials from Quarren Criminal Investigations to abduct aliens to relieve a massive labor shortage. This also included altering their memories so the victims would not realize their situation, and would actually be happy to be working on Quarra. Voyager personnel were victims of this conspiracy, as were numerous others. After being discovered, the Quarren ambassador promised that all the victims would be freed. (VOY: "Workforce", "Workforce, Part II")
In an alternate 2390, Captain Geordi La Forge told the renegade Chakotay that charges of conspiracy to violate the Temporal Prime Directive would be dropped if he, Harry Kim, and Tessa Omond surrendered themselves and a Borg temporal transmitter in their possession. Tessa pointed out that such an offer had little meaning to them, as, if their plan to prevent the USS Voyager from crashing in the Takara sector was successful, the timeline would be changed and the charges would not exist. (VOY: "Timeless")
In 2399, Raffaela Musiker stated to Jean-Luc Picard that she had concrete evidence that a high-ranking Stafleet official conspired to allow the Attack on Mars by rogue synths to go forward. Picard was doubtful, believing that the Romulans had no reason to sabotage their own rescue effort, but admitted he needed her mind to see things that others didn't see, as he investigated what he believed to be the presence of secret Romulan assassins on Earth. (PIC: "The End is the Beginning")
- Main article: Khitomer conspiracy
The Khitomer conspiracy was a plot in 2293 to sabotage peace talks between the Federation and the Klingon Empire by assassinating Chancellor Gorkon as well as the Federation President, and framing the opposite side for the murders.
- Main article: Militarization conspiracy
From 2258 of the alternate reality, Section 31, led by Admiral Alexander Marcus, conspired to militarize Starfleet for war against the Klingon Empire. Marcus recruited the war criminal Khan Noonien Singh, under the name of John Harrison, to design ships and weapons. Khan planned to smuggle out his fellow Augments in a series of advanced long-range torpedoes. However, he fled by the following year, when his plan was exposed, leading him to commit vengeful attacks on the Kelvin Memorial Archive and Starfleet Headquarters.
Marcus attempted to cover up the conspiracy by ordering James T. Kirk to execute "Harrison" with the photon torpedoes he had attempted to smuggle his crew in. However, Kirk learned the truth when he chose to arrest Khan instead of bombing him from orbit. Marcus came in the USS Vengeance, and, when it became clear Kirk wanted to bring Khan to trial and expose the conspiracy, he attempted to destroy the USS Enterprise, but was prevented from doing so. (Star Trek Into Darkness)