Continuity mistake: 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.
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 its 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 mistake: 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).
Continuity mistake: Spock's funeral takes place in Torpedo Bay 1, on the starboard side of the ship. The interior shots, however, show Torpedo Bay 2 (you can see the label and the airlock from the beginning of the movie). Making matters worse, Torpedo Bay 2 was destroyed in the battle with USS Reliant.
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.
Continuity mistake: 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.
Continuity mistake: After the Reliant attack, when Scotty's nephew is talking to Kirk in sickbay, he asks Kirk if the word is given, and Kirk takes his bloody hand then places it against his tunic. After he dies, the next scene shows the bloody handprint in a totally different spot on the tunic.
Continuity mistake: When Kirk and McCoy are in the turbolift, just before Kirk is told about the message from Carol Marcus, they are both standing right next to the door, but in the close up of McCoy, right after Kirk is informed of the message, McCoy has suddenly moved along the wall.
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 mistake: 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 mistake: 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.
Continuity mistake: When McCoy bumps into a hanging body's arm and yells 'Jim.', the next shot shows Kirk lowering a body from an upper level by rope. To his right, another body dangles (feet point up, frontside visible). In the following close-ups of Kirk, this body is much closer to him and is turned over (feet point down, backside visible). No time compression occurs between the two shots. (Director's cut DVD).
Other mistake: During many of the scenes of the Reliant in space, you can see stars through the dark parts of the engine nacelles.
Other mistake: When Captain Terrell shoots himself with the phaser, the phaser disintegrates along with his body, when it should have just dropped to the ground (as often shown in the various Trek TV shows). It wouldn't make sense to explain that the phaser disintegrated because Terrell was touching it, because then it would be impossible to explain why nothing happens to the ground that people are standing on when they get shot (Director's Cut).
Revealing mistake: When Chekhov and Terrell see the Ceti Eels moving around in the tank for the first time, it's obvious that it's not sand in the tank. As the creatures move underneath, all the 'grains' move in unison, and not a single grain of sand is disturbed nor do they tumble over one another.