Goofs

As the camera pulls back after Kirk shoots the Nibiran creature and the native Nibirans reach the spot, the creature is gone. A native even runs over the spot where the creature should have been.

Sulu and Spock's safety belts disappear and then reappear during the final battle sequence.

When Kirk, Spock, and the two security guys come running off the shuttle firing at the Klingons, the two security guys don't appear to be shot, but they disappear completely from the sequence.

When Bones is scanning Kirk as they first get on the ship, he has a ring on his pinky finger; the second time his hand is seen, the ring is missing, but in the next shot, he is again wearing it.

McCoy is performing experiments on a dead tribble. Later, after many tremors, explosions, and the Enterprise being literally upside down, the tribble is seen resting on the same spot. There is no way it could be still on the table, and McCoy obviously had very different problems than care about a dead tribble to put it back there.

When the Enterprise loses power and goes into free fall, the internal gravity also fails. At this point, everyone and everything not held down should accelerating at the same rate as the ship and, therefore, should be floating inside and would not be able to run around the ship.

When Kirk and Khan are blown out of the Enterprise garbage chute, it's done by the ~14.7psi air pressure in the hull rushing through the large port carrying them out. When they arrive at the Vengeance, the port is roughly the same size, and the ship's air pressure would be the same. So as they meet this air stream blowing opposite their motion, they should have slowed at the same rate as they were accelerated before, and fallen to the floor at roughly the same distance inside as when they crouched in the Enterprise. If Scotty left the port open any longer, they'd have been blown back out into space, but they certainly wouldn't glide along like paper airplanes in the Vengeance's artificial gravity, which would affect them the instant they passed into its hull. And someone in the Vengeance a hundred feet from the port in a cavernous room wouldn't feel nearly as much wind as there was in the Enterprise's tube.

As the Enterprise falls to Earth, it spirals down like an airplane. But there is no air to create drag and cause the spin. Nor does the ship have any available propulsion to cause a spin. When it reaches the atmosphere, the spin has somehow already stopped.

As Kirk and Khan travel between ships, the puffs from their suit thrusters fall away behind them like smoke in wind. But there is no air in space for them to fly through or to blow the puffs back. The puffs should shoot out straight, the same as if the thrusters were stationary.

After the Enterprise's flight from Admiral Marcus is halted, Hikaru Sulu states the Enterprise is 237,000 km from Earth; however, the Earth's moon is adjacent to the Enterprise at this time. The moon's distance from Earth is, on average, 238,000 miles, which is approximately 384,000 km.

Addition: In the German dub Sulu says that they're 20.000 km away from the moon. Not sure if that was an attempt to "fix" it or if that was a wrong translation.

If the Vengeance and Enterprise were both traveling at Warp, the Vengeance shouldn't have been able to fire on them with phasers. Photon torpedoes are stated to travel faster than light but not phaser beams. But according to warp travel theory, ships travel within a warp bubble. Therefore, space is traveling faster than light around the ship, but within the warp bubble, the ship is static...not traveling at all. Therefore, since the Vengeance waits until she is extremely close to the Enterprise before firing phasers, it's possible that the two warp bubbles have overlapped. Thus, both ships are static in the same space, therefore the phasers would work as shown.

When the shuttle with Kirk and Spock on board arrives at the Enterprise, the camera angle makes the ship's secondary hull look perhaps twice as wide as the shuttle. But in multiple shots throughout the movie, the interior of the engineering section is considerably larger, so the camera must have been much closer to the shuttle than to the Enterprise.

When Kirk orders Scotty to stun Khan with his standard issue phaser, Khan is dropped to the floor immediately and temporarily immobilized. Later when Uhura shoots Khan repeatedly during the fight with Spock, Khan is barely affected. One might assume that with Khan's engineered superior physiology, the amount of adrenaline in his system from the fight could boost his endurance.

When Khan is describing the dreadnought class star ship, he say's it's 2 times larger and 3 times faster (than the Enterprise). That would mean it is capable of warp 27. Warp 13 is the fastest possible "warp speed" mentioned in the history of Star Trek. However, warp speed whole numbers increase exponentially. According to The Star Trek Encyclopedia, a warp speed calculation of the era suggests "3 times faster" is about warp 13. Beginning with TNG, warp speed calculations were updated to scale differently, but the same limitations still applied. Only the numbers changed. Later calculations placed warp 9.99 as the fastest before entering "transwarp", while previously, and in the era of this movie, warp 21 was considered "transwarp".

(at around 1h 24 mins) While planning the space jump, Sulu's display incorrectly labels the Enterprise as NCC/0514, which is the registry for the USS Kelvin from Star Trek (2009). It should read NCC/1701.

In several scenes, the seat belts on board the Enterprise are clearly not attached to anything.

When the officer from the hospital is given Khan's blood in a laboratory vial, it acts as if it was water, something that blood can't do since it leaves a diluted trail.

Khan states that the Enterprise's life support is located "behind the aft nacelle". This is meaningless. There are two nacelles but they are in line with each other. Moreover, the nacelles are the rearmost parts of the ship: there is nothing behind them.

During Kirk's speech to the crew what appears to be the monorail (No. 234) safe working load limit of 1000 lbs is shown in the background. US/imperial measurements were obsolete before the Federation was created; only standard international units are used.

The establishing shot of London in the beginning of the movie, just after the main title, shows the morning sun rising over the city's skyline from the west.

When the Enterprise reaches the Klingon home world, the title card mistakenly identifies it as "Kronos" rather than "Qo'noS."

During the finale, Khan needs Spock to lower the Enterprise's shields so he can beam the 72 torpedoes containing his crew onto the Vengeance. However, earlier, General Marcus beams his daughter Carol onto the Vengeance without needing the Enterprise to lower its shields.

Even if no one else could figure out how to use Scotty's trans-warp beaming equation to beam Khan back from Qo'noS (given that neither planet is at warp anyway), the Klingons could certainly beam a larger force into the abandoned city on their home planet, and any transporter on Earth could have assisted the Enterprise when it was near the Moon.

After the reveal that Harrison is actually Khan, during which only a few characters are present, Kirk returns to the bridge and instructs security to put Khan in the brig, but no-one on the bridge should be aware that it's Khan at this point.

When about to begin their "never before attempted" five-year mission, Kirk asks where they should go. It is absurd to think that Star Fleet would not have spent a very long time planning this. Also, based on Kirk's unreliability, his lying on official reports, his initial willingness to violate ethics with the intention of willfully murdering a villain, and particularly because of his disregard for the prime directive, there is no chance that he would have been given this mission. If this were to happen today in, say the Navy, there is no way that he ever again would be given a command.

Cyborg science officer "0718" is asked by Kirk where the damage is. He responds by saying, "Major hull damage, captain." It is a nonsensical response. It is obvious that the hull is damaged. The character doesn't say where.

When Spock is to beam down after Khan, Checkov gives co-ordinates to the Transporter Room. The Transporter Officer then says; "Co-ordinates confirmed" when he should have said; "Co-ordinates received" as he has no way of confirming the values.