Corrected entry: The movie takes place in the year 2259. Every reference to Khan's hibernation stated that he was frozen for "300 years", which would mean that genetically engineered superhumans were created in the year 1959. Naturally, there's some leeway when someone refers to "300 years" but even setting the threshold to 250 results in the year 2009. It would have been more convincing if they had set it at "200" or "over 200 years".
Corrected entry: *Spoilers* - Shortly after Kirk is killed by the radiation in the warp core, Dr. McCoy realizes that Kirk's cells can be regenerated by Kahn's blood. McCoy desperately tries to signal Spock to tell him not to kill Kahn because they need his blood to save Kirk. Apparently McCoy totally forget that they have 72 other genetically-enhanced members of Kahn's crew on board the Enterprise to harvest blood from. He even removes one of Kahn's crew from their cryo-stasis chamber so that he can use it to keep Kirk's brain cells viable; he actually had a genetically-enhanced man right there on a bed in sickbay to harvest blood from.
Correction: Khan's crew were frozen in cryo sleep for years. Removing them from cryo sleep willy nilly like that would most likely cause them to die. McCoy even states this shortly after the discovery of Khan's crew. Blood from a living member of Khan's crew stands a better chance to save Kirk's life and frankly the time it would take to unfreeze and revive a crew member would be extensive and time that they simply don't have.
Corrected entry: When Kirk gives Sulu the con before leaving for Kronos, Sulu states "I just never sat in the chair before," referring to the Captain's chair. Sulu commanded the Enterprise when Kirk and Spock beamed aboard Nero's ship to save Earth and Pike. (00:47:50)
Corrected entry: Khan and his 72 counterparts were all a breed of the same genetic engineering. Why would McCoy remove the guy from the cryotube to keep him in a coma so Kirk's body could be saved for Khan's blood? The other 72 super humans were identical to Khan. No need to save Khan for his blood when there were 72 other donors right there in sickbay.
Correction: Dr. McCoy had only tested Khan's blood and he wouldn't have known if any other Augments' blood would have worked. Testing their blood would take too long, but he still needed a backup plan. But the surest way of saving Kirk was with Khan's blood. Details of this are explained in the novelization of the film, but even without this explanation, there's no plot hole. It should also be pointed out that Spock was the one who convinced Kirk not to kill Harrison, so logically he shouldn't have wanted to kill Khan, just emotionally as Kirk had wanted to.
Corrected entry: The Vengeance may have been an advanced model of starship, but it was ultimately made with comparable materials as other ships. Having 72 torpedoes detonate within the hull should have (or nearly) destroyed the entire ship instead of just blowing a hole in the side of its hull. The strength of its advanced shields would be irrelevant from the inside of the ship. (01:39:25)
Correction: They are not 'full strength' torpedoes. It is revealed that those devices are nothing more than containers of the ancient incubation chambers. Each with a small explosive device designed to destroy its contents and not do major damage to external property.
Corrected entry: After the Vengeance has crashed into Earth, Spock is chasing Khan and at one point they run across a street and we see a car brake suddenly to avoid them. When the camera view changes to a street level view we can see some pedestrians walking past the car and they can be seen passing through the front of it, showing that it was added using CG.
Correction: It is very clear that these people walk past in front of the vehicle. Further evidence (now) shown in the making-of's show that a vehicle wrapped in green material was actually used in these shots, so the only part of the car replaced would be an area they'd avoided in real life anyway.
Corrected entry: In the final fight between Spock and Khan, you see them hitting each other on top of a ship/shuttle, and then one after the other, both jump on top of another ship/shuttle passing below. When Spock jumps, on landing he rolls backwards and almost falls off the shuttle. When climbing back up, Khan realizes this and starts running towards Spock to kick him, which he does. In the next shot, Spock is thrown back from the kick, however he is not on the ship's edge but several feet towards its middle, with no time to have got there.
Correction: Watch the scene carefully. Spock pulls himself up and is on his knees when Khan notices him. Khan starts to run towards Spock. Spock quickly starts to crawl towards Khan. They meet towards the middle when Khan kicks Spock. Spock partially blocks it with his arms, but still gets knocked back. When they show Spock get thrown back, he's exactly where he should be on the shuttle.
Corrected entry: As soon as the Vengeance's weapons fail to destroy the Enterprise, Scotty was revealed to have gone to engineering and turned off the shields. Scotty then phones Kirk to beam him back to the Enterprise and Kirk points out the Enterprise is low on power. Scotty then replies "What do you mean? Low on power? What happened to the Enterprise?" How did Scotty not know that the Vengeance damaged the Enterprise? As the Enterprise and Vengeance were fighting in mid-warp earlier on, the Vengeance kept blasting the Enterprise, knocking it out of warp. If Scotty was in engineering the entire time, he should have been aware of the fact that the Vengeance's weapons were currently in use whilst the Enterprise was right next to it. (01:20:50)
Correction: Scotty was hiding the whole time. So he wasn't out in the open, looking at the monitors to determine what was going on. Since no one fired on the Vengeance, he would not have been able to tell that there was a battle going on. He didn't come out of hiding until after the Enterprise was crippled by the Vengeance.
Correction: The 300 years is accurate, because this movie does not take place in our universe. There is a significantly different history between the Star Trek universe and ours, including time that has already passed for us. In the original series Khan's crew was frozen for 300 years (give or take), and this had to be kept the same for continuity reasons. More specifically, it is explicitly stated in The Wrath of Khan that Khan and his followers left Earth in 1996. That puts it at 260 years. One simple rounding job away from 300.