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When the debris falls on Joachim.
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When the mind-control worm is phasered into nonexistence, its squeal can still be heard after it has disappeared.
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As McCoy and Scotty are preventing Kirk from entering the irradiated chamber with Spock ("You'll flood the whole compartment!"), when the camera is on Kirk from the front, they're facing in the same direction as Kirk, but when the shot switches to a POV behind Kirk, they're both facing the other way.
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During the docking with the Enterprise, shuttle control gives them permission to dock at the torpedo launching bay. Instead, the shuttle is clearly shown to dock near main engineering. In addition, Kirk and Co. clearly board the ship through the torpedo bay. The shuttlepod docking sequence, as well as additional visuals of Enterprise in and departing from drydock, are reused footage from Star Trek: The Motion Picture.
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Khan has a warning torpedo fired at the Enterprise. After the torpedo explodes we see David and Carol Marcus, and McCoy, in the sickbay reacting to the explosion. Only six seconds pass (of movie time as well as "real" time) before the turbolift doors open on the bridge with David as a passenger. In the Director's Edition DVD, restored footage allowed a longer period of time between Khan's "warning shot" and David's appearance on the Bridge a moment later.
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During the first battle, the blasts from the Enterprise's phasers hit the blue dome at the rear of Reliant and it explodes. In an aft shot of Reliant moments later there is a slight shot of the dome still intact, though not brightly lit.
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In an overhead shot of Space Station Regula 1 over a moon (directly after the Enterprise departs from drydock), Regula 1 is noticeably transparent; craters can be easily seen through the station. This is seen as a goof, but is actually a limitation of the blue screen shots movie makers faced in the seventies and eighties.
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When Khan leans forward, the wound on his chest "crinkles" visibly, as only a glued-on rubber prosthetic would do. However, his wound seems to be fresh and/or infected - causing the skin to swell. If this wound is now compressed by the surrounding skin/muscle tissue, it will wrinkle as seen on screen.
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Khan says to Chekov, "And you - I never forget a face. Chekov, isn't it?". Although Chekov was not a bridge officer in the TV show that first featured Khan, it should be remembered that when Khan first took over Enterprise, he started with the engineering deck. Chekov was engineering ensign at the time, according to the movie's novelization.
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Khan tells Kirk of the "old Klingon proverb", that "revenge is a dish best served cold". This is, in fact, an Earth proverb.The actual Klingon proverb that Kahn uses to illustrate the Earth proverb on revenge is "It is very cold in space".
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Shortly after the Enterprise and Reliant enter the Nebula, there is an overhead shot of the Enterprise with Reliant down below in the distance. During this shot, the rear-end of the Enterprise's starboard nacelle disappears for a few seconds.
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In the computer image of the Genesis probe approaching the planet, the camera is about to collide with a mountain when a narrow canyon blinks into existence to let it through. During a talk at SIGGRAPH shortly after the movie's release, one of the animators of the clip showed the clip and explained that (at one hour/frame being rendered on a VAX 11/780), they'd rendered every 64th frame of their fractal landscape, checked for obvious issues, then rendered every 32nd frame that was not already rendered. They continued down to every 16th, 8th, 4th, and then every remaining frame. It was only at that point that they discovered that four frames were inside a mountain, and they didn't have the time to rerender with a different trajectory or a different initial landscape, so they papered over the mistake with the magically-appearing valley.
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In the Kobayashi Maru scene, the images the Klingon ships on the viewer are from Star Trek: The Motion Picture. The image of the Klingon Ships firing are from when the Klingon Ships were firing on V'Ger.
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