When the crew is pulling up the grates to lower the photon torpedo into its launch track, you can see a crewman in a red jacket in the background. The way he is fidgeting, he is obviously an extra that didn't know what to do after "Action" was yelled. Though, I guess we could be charitable and say he was a green ensign unsure of what to do at Red Alert. See more...
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Continuity: When Khan's number one officer is dying in the arms of Khan, they exchange a word or two before the first officer dies quite dramatically with his eyes open. Khan then fully embraces the corpse and looks up to the viewer screen and vows to get even with Kirk. However, the "corpse", whose eyes are open, closes them upon Khan's embrace.
Continuity: When Reliant is hit, a large piece of debris falls to the deck, killing Joachim. In the wide shot as Khan makes his way to the debris, a large mass of hanging wires dangles from the ceiling, lower than Khan's head while he is bent over. But the next closer shot has Khan standing erect, lifting the debris, and the wires have vanished. (DVD Director's Cut).
Audio problem: When Kirk says, "Stop energizers," watch his lips closely - he really says, "Stop engines." Mike Okuda points this out in the text commentary on the Director's Edition DVD. The reason the line (and also the one following it) was redubbed was because someone pointed out during post-production that the engines couldn't simply be stopped on a dime.
Character mistake: While Kahn is "interviewing" Chekhov and Terrell, he stated, "On Earth, two hundred years ago, I was a prince, with power over millions." The official date for this movie is 2285. That would place Kahn on earth around 2085 by this statement. However, it is made clear in the episode "Space Seed" that Kahn and his followers escaped earth in the year 1996: nearly one hundred years earlier. Quite a way off to be a rounding error.
Continuity: When Spock is dying within the glass confines of the ships nuclear power source room with Admiral Kirk (William Shatner) on the opposite side of the glass, in one scene speaking to Spock, Admiral Kirk's red Federation uniform jacket lapel is unbuttoned at the top. In the very next scene the uniform jacket is buttoned.
Plot hole: Why doesn't Reliant know that Khan is exiled here? The Federation is so terrified and opposed to genetic engineering that it's still illegal 300 years after Khan. So why is there no warning along with the data on the Ceti Alpha system? Kirk logged what happened with Khan and his solution of marooning him. Starships use nav data to navigate star systems. Ceti Alpha 6 exploded, yet the helmsman or computer never noticed that there is one less planet than there was when Kirk was there? There is no debris from the explosion? Ceti Alpha 5 is the exact same size and was conveniently blown into the exact same orbit as Ceti Alpha 6 used to have? So there is nothing whatsoever to make the crew even suspect that it's not 6? Enterprise would have to have scanned the planets in the system to know that one was habitable for Khan. Did Ceti Alpha 6's destruction somehow magically turn Ceti Alpha 5 into it's exact duplicate? If Starfleet ships have been there to map after Ceti Alpha 6 exploded, none of them bothered to check on the exiles? Pretty callous for Starfleet, don't you think? With the technology and amounts of information available to Starfleet vessels, there is NO logical reason for the Reliant to think that this planet is Ceti Alpha 6. Finally, would the Federation be willing to test a device whose exact effects will be unknown on a planet so close to another inhabited one?
Continuity: After the reliant explodes and the Enterprise is safe, Kirk goes to engineering to see Spock in the radiation filled chamber. During this entire scene, Kirk's tunic is open as Scotty and McCoy keep him from entering the chamber. As Kirk approaches the chamber to talk to Spock, the tunic is partially closed. The next shot shows the tunic all the way open as Spock walks towards Kirk.
Continuity: After being marooned inside Regula I, McCoy is tending to Chekov who is lying unconscious. He says, "He's coming around." You then see McCoy handing him a bandage to hold to his ear and Chekov raises his arm to hold it. Immediately following, there is a shot of McCoy saying "...She's finished," (referring to the Enterprise) and you can see Chekov raising his hand to his ear again.