Continuity mistake: Admiral Marcus' ship "The Vengeance" comes out of warp and immediately appears to be right beside the Enterprise. The top of the ship even goes over the roof of the Enterprise to prove it. Yet from an immediate side angle of the two ships, the distance between the two ships has increased noticeably and the Enterprise is now facing towards the Vengeance.
Continuity mistake: Just before Spock and Kirk go to seek out Harrison, Kirk is examined by Bones. During the examination Bones is wearing a pinky ring, it's then gone, and then back again.
Factual error: After the USS Vengeance blasts the USS Enterprise out of Warp, Sulu says that they are 237,000km away from Earth, but the Moon is between both ships and Earth. Earth's Moon is about 380,000km away from our planet, so the Moon should have been way behind the combatants.
Factual error: At one point Khan threatens to target the Enterprise's life support systems which are located behind her "aft nacelle". The Enterprise doesn't have an aft nacelle; as the many exterior shots show, it has a port nacelle and a starboard nacelle. Khan a) is familiar enough with Starfleet ships to know this basic fact about the Enterprise, and b) can see its nacelles for himself on his own viewscreen while he's delivering this line. There is no "nacelle" housing the impulse engines. They are enclosed by the hull of the saucer section.
Continuity mistake: As Kirk and Spock converge in the lobby of the Daystrom building, Spock greets Kirk with "Captain" and Kirk responds "Not anymore Spock, First Officer." There is a cut between "Not anymore" and "Spock." During the cut Spock moves from being in front of Kirk to behind him and the two have moved from the middle of the lobby, with the ramps visible 10 to 15 yards ahead of them to right beside the ramps.
Factual error: During the Vengeance's fall to Earth, just after hitting Alcatraz, the saucer section hits the surface of San Francisco Bay, the outer rim diving dozens of meters into the water and its rear lifting up almost vertically into the air. A few seconds later the ship is seen from the onshore view, and it is straight and level and heading directly at Star Fleet Headquarters. Hitting the water at that angle and speed, so close to the shore, the ship could not have righted itself and gotten on that trajectory going that fast. It should have sunk or flipped over when it hit the water. Even if it could flip 360°, there was not enough time, distance, or height for the ship to continue on to its target at that speed. The water's drag would have slowed it too much.
Continuity mistake: After Harrison (Khan) has attacked the senior officers' meeting and Pike has died, the position of Pike's head on the cushion changes between shots. When Spock disconnects from the mind meld, the stitching on the cushion is above Pike's ears. When Kirk checks for a pulse, the stitching is below Pike's ears. The angle of his head also changes.
Continuity mistake: In the bar, Pike meets up with Kirk to tell him that they gave him the Enterprise. When Pike tells Kirk that Spock has been reassigned to the Bradbury, Kirk pours liquor into his glass, nearly covering the big round ice cube. When Kirk lifts the glass from the bar, half of the liquor instantly disappears.
Factual error: Enterprise and Vengeance come out of warp near the moon, 237,000km from Earth. Due to their altercation they lose power and proceed to fall freely toward Earth. The scene plays as though it takes a matter of a few minutes, too fast to get the situation under control. In reality the freefall time from that height to sea level is about 13 days and 6 hours; plenty of time to effect repairs or call port authorities for aid.