The one plot hole seen in this movie is in the battle between the Enterprise and Vengeance. The battle forces the Enterprise to drop out of warp, 237,000 KM from Planet Earth. If the Vengeance hadn’t damaged the Enterprise in the first place, then when were they planning to drop out of Warp? At light speed, if they had travelled one more second, it would have taken the Enterprise way past Earth.
Continuity error spotted in the bar scene. Pike meets up with Kirk and tells him that they have given him the Enterprise. Then, when Pike tells Kirk that Spock has been reassigned to the Bradbury, Kirk pours some alcohol into his glass, almost covering the ice cube. When Kirk then lifts the glass off of the bar, half of the alcohol instantly vanishes.
Another continuity error is when Admiral Marcus’ ship comes out of warp and appears in front of the Enterprise. You can see set of panels begins to close over the main deflector dish. Looking out of the Enterprise’s viewing screen in the shot, the deflector dish is now fully visible again.
A revealing mistake found, was that after the Vengeance has crashed into Planet Earth, Khan is being chased by Spock and at one point, they run across a street. We then see a car brake suddenly, to avoid them. When the camera changes view though, to a street level view, we see some people walking past the car and can be seen passing through the front, showing that it was added using CG.
Yet another continuity error and a slight spolier too. Once Kirk has been irradiated when realigning the reactor core, Spock visits him at the reactor door, where they place their hands on either side of the glass. Spock then makes the Vulcan salute (“V” sign) which is visible after Kirk’s hand slides down in the shot from Kirk’s side of the glass. The next shot is from the outside on Spock’s side and Spock’s hand remains flat against the glass but no longer in a salute.
When the Enterprise is spinning out of control, a continuity error was exposed as Spock and Sulu activate their seat belts across their shoulders. Sulu’s seat belt disappears shortly afterwards, only to reappear seconds later.
Another continuiry error was spotted in one earlyish scene, when Spock and Uhura are in a turbolift. In this scene, you can see that Zoe Saldana has a faint spot or similar on her upper lip that has been covered up. In scenes following though, there’s no sign of it.
There is one hell of a factual error as the Enterprise and Vengeance come out of warp at 237,000km from Earth. They then lose power and proceed to fall freely towards Earth. This part is made to seem that it only takes a few minutes, which would be way too fast to get the situation under control. If this were to happen in reality though, seeing as the freefall time from that height to sea level is about 13 days and 6 hours, which would be plenty of time to put repairs into action or to call port authorities for help.