I fully agree with Bob here, this movie and the one before it are incredibly stupid. By throwing in all sorts of references to the show, they're trying to appeal to long-time fans, but failing by having everyone act so moronic.
Yes, there have been episodes and movies in the Star Trek franchise that are stupid and moronic (Final Frontier, anyone?) but on the whole, there is a fair degree of intelligence amongst the characters.
SPOILER alert, don't continue to read this if you don't want to spoil the movie.
Below is a list of things that I can recall (it's been 5 days now since I've seen it) that are completely off the wall stupid. As in, if these people were real, and actually did these actions, they would rightly be dragged before some sort of tribunal and discharged.
1)Kirk being captain of the Enteprise. He is far too young and inexperienced to be a captain. He stops one bad guy, and suddenly that means he's fully capable of running a starship of 400+ crewmembers? Of conducting intersteller diplomacy? Shatner-Kirk got the chair at 29, but only after he'd already been an officer and had earned numerous decorations and awards.
2)The Enterprise at the start of the movie is underwater. Why? The Enterprise is a STARShip, as in, a ship that travels through space. No reason is given for why Kirk felt it necessary to submerge his ship, especially given the Prime Directive: he parked it barely a couple miles from a primitive tribe. No, the only reason I can fathom is one that violates cause and effect: Kirk had to park it underwater, so that he could get in trouble for violating the Prime Directive, leading to conflict between him and Admiral Pike, which is unresolved at the time of Pike's death. In other words, the character in universe had no motivation, no reason to do so, only an impetus that comes from beyond the fourth wall.
3)We're told that this new Kirk and Spock are friends, but I haven't seen any evidence of that. In fact, it reminds me a lot of what happened with the Star Wars prequels: Obi-wan in the original trilogy said that he and Anakin were great friends, but the prequels never displayed any evidence of this. The characters say they're friends, but never actually display friendship: they're always bickering and being annoyed with each other.
4)Scotty is either the world's greatest infiltrator or Admiral Marcus has the worst security ever. Scotty is told a set of co-ordinates by Kirk, flies out in a small ship (important point to remember here, is that no-one knows about cloaking technology, since the Romulan cloaking device has not yet been shown in the Abrams-verse) and flies in to Marcus's shipyard by employing the ninja technique...of slipping in amongst a line of similar shuttles. Given that this is a secret shipyard, where Starfleet's biggest bad-ass warship is being built, you'd think that at the very least, Scotty would have had to provide some sort of ID upon disembarking. There is no way for Scotty to have gotten past security like this, as he was not told what he was supposed to have been looking for, therefore he wouldn't have had counterfeit ID set up beforehand.
5)The fact Harrison is Khan doesn't actually mean anything. It's used as a reference to a much better movie, meaning that those who know who Khan is will only be constantly reminded of a much better movie, while for those who don't know...it doesn't actually mean anything to them. Also, the time frame given by Khan is unrealistic to say the least. If I recall the details correctly, Khan says he was woken up by Marcus sometime between the previous movie and this one. Darkness takes place a year after the first movie. In that time, Khan was woken up, learned and mastered sciences and technologies centuries ahead of what he was used to and in fact, improved upon them? What type of genius is he? We're told the Enterprise bridge officers are all geniuses: why is it that they, who have lived with this level of technology all their lives and understand it intimately are unable to provide the same level of advancement that Khan has? Imagine if...today da Vinci or Edison or Tesla woke up today. Would you honestly expect them to learn and master all the advancements in science since their deaths and not only that, but to improve on them, all in less than a year?
6)No-one at Starfleet questioned Marcus's orders to launch 72 missiles onto the Klingon Homeworld (by the way, I noticed that it's moon, Praxis, is already in pieces, even though in the original series, that happened in 2293 due to over-mining. Here, there's no explanation given when we see half a moon). No-one wondered why Marcus basically said to provoke a war. The movie implies that amongst the top brass at Starfleet, it was Marcus alone who was plotting the militarization of Starfleet. No-one says in that emergency meeting "Hey, Marcus, why the torpedoes? Isn't a stealth team enough? Do you, the head of Starfleet in a peaceful and democratic federation, want a war with the Klingons?" Imagine if today, a terrorist bomber well bombs a US military base, flees to say Iran and the President says to a ship captain, send in a squad to apprehend him, but if you fail, bomb him. In fact, uber-bomb him, it's just unpopulated territory of a sovereign nation that is just itching for an excuse to declare war, to say we're the bad guys who launched a massive unprovoked attack on an independent nation.
7)Were Marcus's bridge officers all as insane as he was? Did his first officer not wonder why Marcus is deliberately firing on the Enterprise? Did no-one think to stage a mutiny against a man who has just calmly stated he is going to destroy a ship full of loyal Starfleet officers and start a war? I can understand if the officers believed or were convinced that war with the Klingons was inevitable, so might as well build all this weaponry and attack them first, but that's a far cry from also not caring when your commanding officer shoots a sister ship and kills fellow officers and personnel. In fact, Riker's captain on the Pegasus suffered a mutiny when his crew realized that what he was doing could violate a treaty with the Romulans and provoke a war.
8)The battle between Enterprise and the Vengeance takes place in orbit around Earth. That's right, this film has repeated the now legendary mistakes of certain other movies where it seems that the Enterprise is the only ship in Starfleet. No other ships are dispatched to find out just what is going on. No one on the Enterprise even thinks of attempting to contact Starfleet or the Federation Council or the media or anything. If they had, Starfleet would have sent a group of ships, disabled both the Enterprise and Vengeance and taken both Marcus and Kirk in for questioning.
9)McCoy injecting Khan's blood into a dead tribble. Why? Do doctors, upon finding someone with unusual blood, inject it into dead animals as a standard research procedure? Did he know beforehand that Khan's blood can resurrect the dead? If so, why don't more people use such blood?
That's all I can think about off the top of my head.