Continuity mistake: At the end of the film when Kirk is being awarded his medal, behind the Federation Counsel you can see 3 flags in the background- the center flag is the California state flag. After Kirk shakes Pike's hand, and the camera pans up and back, the California flag is hanging significantly different so that it is no longer recognizable, seeming almost blank white. As the ceremony is indoors and all attending are standing motionless, this flag should not have moved.
Continuity mistake: When Kirk is sneaking about on the Romulan ship, just after Spock takes off in the ship, we see a shot of him walking down a dark corridor, carrying what is very clearly a Romulan pistol (quite long and thin). When the shot changes, he suddenly has his standard Federation Phaser in his hand when he confronts Nero. He doesn't get a Romulan pistol like that until after the fight with the first officer, when his phaser falls over the edge.
Continuity mistake: When McCoy brought Kirk onboard a shuttle as a patient, they boarded a shuttle with round nacelles near the bottom. Most of the shots of it in flight show this. But the shot that zooms out from their window while heading to the Enterprise shows they're on a shuttle with wings near the bottom and nacelles at the top corners.
Continuity mistake: When young Kirk is driving the stolen car he unhooks the latches to the convertable roof. Cut to a side view when the roof flies off and we see that the car is kicking up a big cloud of dust from the roadway (so much dust that it looks like they cut from the paved road to a dirt road on the side shot, but the grass obscures the road top). Cut to a camera angle behind the car when the roof lands and the dust cloud has instantly disappeared, there is no trace of it.
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.
Plot hole: Despite their awareness of the red matter and the power of the singularity, no one on the bridge had the common sense to suggest pulling back to a safe distance. Nero had already refused their assistance so there was no need to stay so close to the singularity. And when a black hole was created inside Vulcan, an officer mentioned they had to reach minimum safe distance, so they know what they need to do, but no-one does 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 the USS Kelvin is attacked by the Romulan ship, there is significant damage done to the front half of the saucer section. However, after Ayel appears on the viewscreen demanding the captain, there is no damage or debris on the saucer section visible through the front windows of the bridge.
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).
Revealing mistake: When the Federation fleet is preparing to warp to Vulcan, you'll see the first ship blast off, but the second ship blasts off out of nowhere. You could argue that the ship was out of sight and below the Enterprise, but the light trail isn't that long. The light trail starts in front of the Enterprise where you should be able to see a star-ship.