While Kirk is talking to Pike in a bar he has a nearly empty drink. Then during a shot when he lifts the glass to take a sip, the glass is full. In the next shot, when he actually takes the sip, the glass is nearly empty again.
When the Enterprise's trash exhaust port is first shown, it's about half-way up the secondary hull and has sliding panels on either side of it. Almost two minutes later, the panels are missing and the exhaust port appears to be farther down and forward.
During the Spock/Khan chase scene, when Spock is jumping onto the flying craft he has his laser in his hand, but then when he jumps it is suddenly not there and there is no way for him to put it into his belt shows because he holds on to the craft with two hands.
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 same table and McCoy obviously had very different problems than care about a dead tribble to put it back there.
Just before Kirk drafts Khan to help him board the Vengeance, Spock and Kirk have an intense discussion in the hallway. As Spock is lecturing Kirk about making wise decisions, a female crewman, (in a red uniform, with hair up in a bun, and reading a handheld device), walks by Kirk and Spock in the hallway. A few seconds later, when Kirk yells "You're right!", the same female crewman AGAIN walks past Kirk and Spock in the hallway. (The second time she walks by them, she looks at them as Kirk raises his voice, and appears to walk away shaking her head.)
When Kirk grabs Scotty while hanging from the walkway, his thumb is not wrapped around the bar. When the camera cuts to a close up of Kirk's hand a few seconds later, his thumb is wrapped around the bar. And in the shot immediately following, his thumb is not around the bar.
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.
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 can be seen 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.
When the Enterprise comes out of warp, Sulu states that they are 237,000 Km above the Earth. After Khan's vessel explodes, the Enterprise loses power and begins falling. Even assuming full gravity for so high an altitude, an object would take almost two hours to fall to ground level (Ok, call it 1 hour, 50 minutes to break up in atmosphere.) On screen, this took about 3 minutes.
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 Kirk & 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.
As Kirk is trying to realign the warp core, Sulu declares that if they don't get the core back online, the Enterprise will burn up on reentry. When the core is realigned and power restored, the Enterprise is already within the earth's atmosphere, as evidenced by the clouds. The ship did not, in fact, burn up upon entering the atmosphere.
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.
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.
During the Spock/Khan chase scene, Khan sprints across a busy street and causes a car to stop in the middle of the crosswalk. As Spock chases behind him, you can see four pedestrians walking through the front of this car.
The goof items below may give away important plot points.
During the exchange negotiations between Khan and Spock, Khan says he will destroy the Enterprise's life support, located behind the aft nacelle. The nacelles are the two tube-like structures behind the saucer section, one to the right and the other to the left. There is no 'aft' nacelle. Also considering the nacelles protrude out behind the ship, there is nothing behind them.
When the Starship Vengeance is approaching San Francisco (when you see San Francisco in the background), the sun is in front of the ship, however when the ship hits the water after destroying Alcatraz, the sun is now behind it.
When Kirk grabs Scotty from the walkway, there is clearly no one nearby, and gravity is clearly sideways from the railing. So there would be no way for Chekov to appear in time to grab Kirk's hand, even if he had been working his way along the railing.
When Scotty resets the Vengeance's systems, the large ventral weapons suddenly droop as if pulled down by gravity. But there is no gravity, or the debris between the ships would have fallen away. It would require not only power, but a command to point the weapons down, and the story at that moment is that no commands can be given to the weapons.
During the final battle, the Enterprise is in orbit adjacent to the moon. At that time its drive is disabled (no warp, no impulse, no thrusters) and falls to the Earth in a few minutes. Any object that close to the moon would have landed on the moon's surface.
After the final battle with the Dreadnought class ship, when the Enterprise's engines fail, it starts dropping towards Earth. However, during the Battle both ships and the debris from the battle are clearly in a stable orbit which does not require power to maintain.
It makes no sense for pedestrian traffic and utility transport service in the city to behave as though it was a normal day moments after a large spacecraft crashed into the city and still looms higher above the ground than most buildings, in view of anyone on the ground. The impact alone would have felt like an earthquake, which would certainly get everyone's attention in San Francisco.
After Kirk dies from radiation poisoning, much ado is made about Kahn being the only chance they have to bring him back to life by using Kahn's "super blood." While this allows for a dramatic, action packed chase scene; the obvious solution would be to use one of Kahn's crew to synthesize the serum since they were all subject to the same Eugenics experiments (according to the Star Trek mythology). One would imagine at least 1 of the other 72 crew had similar platelets.