Continuity mistake: During the scene in which Kirk first meets McCoy in the shuttle, watch the safety harness strap over McCoy's right shoulder. As the perspective changes back and forth from McCoy to Kirk, the strap is bunched up, then flat, then bunched and then flat again without McCoy touching it.
Continuity mistake: When Kirk, Sulu and Olson are space jumping down to the platform, they are shown on camera as falling with Olson in last place. He is furthest from the platform. Yet on Chekov's screen, he is displayed (by the red color associated with his red uniform) as being the lowest and closest, in first place.
Continuity mistake: When young Spock approaches the three bullies at school, the shot from behind Spock shows he is at least as tall if not slightly taller than the other three boys. But when the shot reverses to see Spock between two of the bullies, he is instantly much shorter than they are. This isn't a matter of camera angle, but rather the choice to have young Spock positioned lower to make the bullies more intimidating - a choice the earlier angle didn't take into consideration.
Continuity mistake: After their discussion of younger Spock's emotional compromise, Spock Prime stays in the same spot while Kirk goes to the transporter pad. They are still talking as they do. Shots from afar show Spock is illuminated by the blue lights over his head as is the walls around him. But each time it cuts to a close up, he and the walls no longer show any hint of this blue. He also appears to be standing in a different location in the close ups.
Continuity mistake: When Spock points out to Sulu that he has forgotten to disengage the external inertial dampener, we see a close shot of Sulu as he corrects the situation on his controls, and Pike can be seen sitting behind him with arms by his side. It then cuts to a long shot where Pike suddenly has his arms crossed.
Continuity mistake: When Nero is (lying down? I can't tell during this scene) and waiting for Spock to arrive, we can see that he has a very faint tattoo of some kind beneath his lower lip. However, when he mentions to his first officer that he doesn't want Spock dead and wants him to watch Vulcan being destroyed, the tattoo has become black and very dark and also extends onto his lower lip.
Continuity mistake: As Kirk is being choked by a Romulan, he says "I've got your gun", then pulls the weapon and kills the Romulan. As the Romulan falls, Kirk grabs the edge of the platform and begins to pull himself up onto it. From well above, we see the Romulan's gun is lying on the platform pointing directly away from Kirk. But when the camera angle changes to show Kirk's face with the pistol between him and the camera, the gun now points to Kirk's left (our right).
Continuity mistake: As Kirk and Spock Prime are arguing about Spock meeting his younger self, the shots from behind Spock show his hair, ear, and coat to be illuminated by a blue light. But the reverse angles have no evidence of this light at all. And when Kirk walks away, Spock is still standing in the same spot. Now suddenly Spock is all illuminated by blue lights that they were obviously not standing under moments before.
Continuity mistake: Just after Nero's ship goes into warp speed to chase after Spock, the scene cuts to Kirk being confronted by one of Nero's men. Kirk jumps off the ledge of the platform and lands on another ledge. His arms grab onto the ledge with his arms stretched out. The next shot shows his arms now next to his body. He then stretches out his right arm to try and climb on some more. The next shot shows both arms stretched out.