Continuity mistake: In the training simulator at the beginning of the film, after Sulu and McCoy have both fallen, McCoy has his head resting on Sulu's hip at first. Then, in the next shot, his head is resting closer to Sulu's knee. (00:05:50)
Factual error: Inside the nebula, the Enterprise rises from below and fires a photon torpedo at the Reliant, striking it directly on the dorsal torpedo pod. Watch as the torpedo hits and the pod explodes. Large pieces of the bulkhead explode outward, then begin to fall downward towards the saucer section before they cut camera angles, despite there being nothing dragging them in that direction. (01:31:25)
Continuity mistake: The Enterprise is (slowly) attempting to put distance between herself and Reliant (which is about to explode). Whenever Enterprise is shown coming toward camera (or watching Reliant on its viewscreen), Reliant is alternately seen from its starboard side pointing at our 2 o'clock position, from port/aft listing hard to starboard, from it's aft/top area, pointing toward 1 o'clock, and from starboard/bow pointing to 5 o'clock... yet Reliant isn't wobbling or moving, and Enterprise maintains a straight (shortest) course away from her... all suggesting Enterprise is traveling in four directions at the same time. (Director's cut DVD). (01:36:50)
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). (01:11:20)
Continuity mistake: When the Enterprise leaves spacedock, Saavik orders 'one quarter impulse power'. However, in the exterior shots the impulse engines are not lit. The reason is that ST II uses stock footage from ST I for the spacedock scene. In that movie, the ship leaves the dock using thrusters. The obvious give away is the blue deflector from ST II forward, while ST I had a red deflector. It is a short scene less than 2 seconds, you really have to watch for it. The thrusters are on, as seen in the parts of the scene with the blue deflector, and suddenly they are off and the deflector is red for a few seconds.
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.
Plot hole: Khan said that the crew of the Botany Bay had been stranded on Ceti Alpha VI for 15 years. Considering that most of his crew were in their 20s, wouldn't that mean they were all children when they were first stranded? In the original TV series, all of these superhumans were already adults, so how come Khan was the only person to age? (00:19:40)
Continuity mistake: In the scene when Khan first opens fire on the Enterprise, the exterior shots show the damage on the forward part of the engineering hull. However, the damage on the interior shots is occurring on the aft section. Which is to say if the exterior and interior damage lined up the way they're supposed to, Scotty and his cadet engineers would've all been killed in a warp core breach. (00:50:15)
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. (01:39:55)
Other mistake: After the Enterprise blows the port nacelle off the Reliant, the destroyed Reliant bridge is shown. The three sheaves of wires hanging from the ceiling should be on a tilt with the angle of the ship rather than straight down. Even accounting for the artificial gravity unit, which could have been damaged.