The scene where Spock is attempting to interrogate Kirk and Scotty, who have arrived on the ship mid-warp.
Spock: [To Scotty] Are you in Starfleet?
Scotty: [Soaking wet] I, um, yes — can I get a towel, please?
Spock: I order you to explain me how you were able to beam aboard this ship while moving at warp.
Kirk: Don't answer him.
Spock: You will answer me.
Scotty: ... I'd rather not take sides.
The towel comment was an ad-lib that the cast thought was so funny, it took all of Zachary Quinto's willpower to not corpse after everyone else had. The final scene that made it in the film has his mouth just twitch enough to pass.
And the best part is that the next time we see Scotty, he has his towel.
"The notion of transwarp beaming is like trying to hit a bullet, with a smaller bullet, whilst wearing a blindfold, riding a horse!"
Kirk's allergic reaction to the vaccine injection.
McCoy: Ah, Jim, you're awake. *looks at Jim's hands* Good God, man!
Kirk: What? *looks at his hands* *hilariously* Eeeh! *raises hands; both are swollen* What the hell is this?
Kirk: ...and wath da sip Womuhln?
Uhura: Was the ship what?
Kirk: Womuhln - *To McCoy* Wath happ'ning to my mouth?
McCoy: You got numb tongue?
Kirk: NUMTON?! *turns to Uhura* Wath the sip Womuhln?
And let's not forget Kirk actually losing the hot lady, in this case Uhura to Spock!
Kirk: So her first name's Nyota?
Spock: I have no comment on the matter. (the way he says it could be Spock's way of saying 'back off my girl')
The best part is Kirk's expression, which basically reads: "WTF!? HOW DID YOU GET A GIRL LIKE HER!?"
Chekov had a few moments : "W-w-v-v-w-victor, W-w-v-v-w-victor", and racing down the hallway yelling "I ken do zat! I ken do zat!" Go, Chekov!
Kirk boards the recruiting shuttle heading for Starfleet...and walks smack dab into a low bar.
Look in the background when Scotty is first introduced. He has a pet Tribble in a cage. No wonder he's so hungry!
Bones as an aviaphobic seatmate:
Pike (voice): This is Captain Pike. We've been cleared for takeoff.
Bones (to Kirk): I may throw up on you.
Kirk: Well, I think these things are pretty safe.
Bones: Don't pander to me, kid! One tiny crack in the hull, and our blood boils in thirteen seconds! Solar flare might crop up, cook us in our seats! And wait till you're sitting pretty with a case of Andorian shingles! See if you're still so relaxed when your eyeballs are bleeding! Space is disease and danger wrapped in darkness and silence!
When Scotty said he'd tested his new transporter theory out on Archer'sbeagle!
"How old are you?" "Sewenteen, sir!" ...aaaaaand cue Bones' exasperated sigh and "Oh good. He's seventeen!"
"Is the parking brake on?"
"Have you disengaged the external inertial dampeners?" "Oh." (Meaning that yes, he did leave the parking brake on!)
This bit in the medbay of the Enterprise:
Bones: I'm going to give you a mild sedative. *injects Kirk with hypospray*
Kirk: How long will this take t- *passes out*
"STOP THAT!!" and the Kobayashi Maru scene. If you watch closely during the opening of the later, you can juuust tell Uhura is being sardonic, and would really like to chew Kirk out.
The Reveal that the person who created the test was Spock!
Spock Prime: "I am Spock." Kirk: "... Bullshit."
What makes it hilarious is that Kirk isn't boggling at the implied time travel, but at the idea that he and Spock could ever be friends.
Then later, when Spock Prime tries to convince Kirk to take command:
Kirk: How, over your dead body?
Spock Prime: ...Preferably not.
Also when Olson (who happened to be wearing red) died, Chekov, in his adorably genuine stunned voice, states that Olson is dead.
Spock to saying "Live Long and Prosper" to the Vulcan Science Academy, you can see the hint of satisfaction in his face and the way he says it, he might as well be flipping them the bird. Not to mention the music for the next scene starts up right at that moment.
Are you out of your Vulcan mind?!?! (From Bones to Spock, with Vulcan pronounced so it sounds like a Precision F-Strike.)
"Hi, Christopher. I'm Nero."
Kirk and his apple during the Kobayashi Maru scene. When he blatantly cheats, he does it with style.