9. I Never Forget A Face
One of the most well-known continuity errors in all of Star Trek takes place in the second movie, The Wrath Of Khan. In fact, it was so well known, that two people spotted it at the time. One was the actor with whom it was concerned. The other was the director, Nicholas Meyer.
When the USS Reliant arrives in orbit of Ceti Alpha V, believing it to be Ceti Alpha VI, Commander Chekov accompanies Captain Terrell to the planet's surface. There, they encounter Khan, the genetically engineered madman who attempted to seize control of the Enterprise fifteen years earlier. Though Khan doesn't recognize Terrell, he does identify Chekov by name. The issue with this is that Chekov wasn't on board the Enterprise during the events of Space Seed.
Walter Koenig spotted this error straight away but, fearing a diminished role in the film, he kept his mouth shut. Nicholas Meyer would also come to realize the error and, years later, said that he could just have easily swapped out Chekov for Uhura in this scene, but he decided against it, leaving it in as a little nod to the fans who noticed.
Off-screen, a joke could be heard describing Chekov as the annoying crewman who held up Khan was the latter was waiting for the bathroom. That's a face we wouldn't forget in a hurry either.