(covers information from several alternate timelines)
This page contains information regarding Star Trek: Lower Decks, and thus may contain spoilers.
Beta Quadrant wreckage
In an alternate timeline in which USS Voyager was lost while using quantum slipstream drive, Starfleet Intelligence recovered a Borg temporal transmitter in the wreckage of a Borg cube in the Beta Quadrant. (VOY: "Timeless")
First contact cube
In 2365 Starfleet encountered the first known Borg cube in System J-25 when Q threw the Enterprise into that system. This event constituted the official first contact, although the Borg themselves had been encountered two hundred years before by the crew of Enterprise NX-01 and also by civilian researchers in the 2350s.
The Borg cube encountered in 2365 dispatched two drones to the USS Enterprise-D, which examined the ship's engineering spaces. The cube then attacked the Enterprise, slicing into the saucer section and using a tractor beam to extract components from three decks. Captain Picard had an away team, consisting of Riker, Data, and Worf, beamed aboard the Borg vessel in order to learn more about the Borg. The away team learned a great deal, but then the Enterprise had to retrieve the team after finding out that the Borg were telepathically using their collective hive mind to repair their damaged vessel. The Enterprise left at warp speed pursued by the Borg cube.
The Enterprise increased to maximum warp, but still had no success in escaping. Riker ordered them to arm photon torpedoes and Picard gave the order to fire, but the torpedoes had no effect. The Borg ship, after getting within firing range, fired a shield-draining missile twice and the Enterprise, with very low shields, fired torpedoes again, with the same result. After determining their weapons were useless, Picard pleaded with Q to send the Enterprise back into Federation space, which he did. It is unknown where the Borg cube went after Q helped the Enterprise escape. (TNG: "Q Who")
Klingon space cube
Sector 001 cube
In 2373, another Borg cube attacked Earth much in the same manner as the previous cube from Wolf 359. The Battle of Sector 001 ended when this Borg cube was destroyed by a Starfleet armada using tactical knowledge of Captain Picard from his time as Locutus. The Borg Queen was on board this cube, and attempted to escape back in time using a Borg sphere. (Star Trek: First Contact)
Nekrit Expanse cube
In 2368, this Borg cube was disabled by an electrokinetic storm within a nebular region of the Delta Quadrant, the Nekrit Expanse. With the majority of the Borg aboard the cube killed, eighty thousand surviving drones (freed from the Collective) fled to a nearby M-class planet. The derelict cube was later discovered by the USS Voyager in 2373 and subsequently boarded by its crew. It was later visited by Commander Chakotay who successfully reactivated the ship's neuroelectric generator, rejoining the surviving Borg into a new collective called the Borg Cooperative. The reactivation of the generator also reactivated the cube and its dormant drones, however the Cooperative ordered the cube to self-destruct before all its systems were fully operational.
The cube's complement of drones consisted of many Alpha and Beta Quadrant species including Humans, Romulans, Cardassians, Klingons, Farn, and Parein. It was the first Borg vessel encountered by the Voyager crew. (VOY: "Unity")
Two Borg cubes
Long-range probe cubes
In 2373, Voyager launched a long-range probe. Several months later, the probe encountered these two cubes and was assimilated. It was this encounter that proved Voyager had entered the heart of Borg space. (VOY: "Scorpion")
Fifteen Borg cubes
In 2373, fifteen Borg cubes passed the USS Voyager in pursuit of Species 8472. One of these cubes stopped momentarily to scan Voyager with a polaron beam. All fifteen vessels were later destroyed by a Species 8472 bio-ship. (VOY: "Scorpion")
Seven of Nine's cube
This Borg cube, containing Seven of Nine, assisted Voyager in developing the bio-molecular warhead against Species 8472 in 2374. When Voyager approached the vessel, it was locked in a tractor beam and Captain Kathryn Janeway was beamed aboard to negotiate the development of the warhead. After Species 8472 destroyed a nearby planet, the cube tractored Voyager to warp and to safety to develop the weapon. Janeway and Tuvok, with Seven of Nine as a representative, worked in the vessel on the weapon until the cube was attacked by a Species 8472 bio-ship. The Borg knew that they could not allow Voyager to be destroyed; and so the cube rammed the bio-ship, destroying both. Portions of the cube were beamed into one of Voyager's cargo bays along with Tuvok, Janeway, Seven of Nine, and a number of Borg. (VOY: "Scorpion", "Scorpion, Part II")
Two Borg cubes accompanying Seven of Nine's cube
As captain Janeway sought to contact the Borg in order to make her proposition, she found that besides Seven of Nine's cube, two other cubes were in orbit around the Borg planet Voyager arrived at. These two cubes were destroyed by the shock wave of the exploding planet, detonated by Species 8472, while trying to protect Seven of Nine's cube from destruction, in which they succeeded to their own detriment. (VOY: "Scorpion")
Borg cubes in fleet
Species 116 homeworld cubes
In 2374, a member of Species 116, Arturis, who blamed Captain Kathryn Janeway for the assimilation of his homeworld, tried to lure the crew of the USS Voyager into a trap. He disguised his ship as a Starfleet vessel, so he could get the crew of Voyager on board. However, his plan was discovered and he only managed to capture Captain Janeway and Seven of Nine before he activated the quantum slipstream drive and set a course for his former homeworld, inside Borg space. Voyager pursued them and managed to rescue Janeway and Seven. As the vessel emerged from slipstream, it was quickly surrounded by at least four cubes. Arturis and his ship were presumably assimilated. (VOY: "Hope and Fear")
Species 10026 homeworld cubes
When Seven of Nine briefly returned to the Borg Collective in 2375, a fleet consisting of the Borg Queen's ship and two cubes left the unicomplex to assimilate Species 10026. Upon arrival they were attacked by 39 vessels armed with pulse weapons, which could penetrate their shields. With the unwilling help of Seven of Nine, the Borg were able to adapt and assimilated almost the entire species. (VOY: "Dark Frontier")
Cube reduced to debris
Battle against the Romulans
These six cubes participated in a battle against eleven Romulan warbirds during the late 2370s. The battle was happened upon by the USS Cerritos before Federation starship hastily departing the battlefield. (Star Trek: Lower Decks opening credits)