"There is a multilegged creature crawling on your shoulder." (I, Mudd) "All right - fine! But I can't wait to get back to Deep Space Nine and see your face when you find out that I never existed!" (Trials and Tribble-ations)
...in TNG, "The Naked Now" when Picard is following Dr. Crusher, and he does that little "skip" thing with the childish smile on his face...cracks me up every time!...
Two I remember from The Next Generation: I enjoyed Riker playing with the Picard doll from the kids' collection of items for Captain Picard Day. Troi telling him to put it back (before he left for the bridge with it) was funny! And it got better, when Picard sort of ended up looking foolish when the Admiral asked about the Captain Picard Day banner. His awkward "I'm a role model.", and the Admiral's deadpan '"I'm sure you are." was great! Oddly, another scene involved Troi (though I don't remember the episode) was regarding Worf's birthday. Troi said something to him about how "he'd probably wish to be alone, and hit himself with a pain stick or something". I actually laughed out loud at that crack of hers!
In "Timescape", Picard imitating Dr. Fassbinder (he...just...kept...talking...in...one...unbroken...sentence) and Troi's impression of the Ktarian Dr. Mizan
weeerrrrrrllll.....A Taste of Armageddon. And yes, it IS funny. It's always the first line I cite when anyone comes out with "Vulcans are incapable of lying." Riiiiiiiight.
Worf and some awkward moments are fun when somebody asks him his opinion and he goes all defensive. And there are many of those. from "All Good Things..." TROI: "Worf, we were strolling barefoot along the beach while balalaika music played in the air... ocean breeze washing over us... stars in the sky... a full moon rising... and the most you can say is "stimulating"?" WORF: "It was... VERY stimulating." Troi walks away WORF: ...what?
In "Generations", where Data says "Open Sesame!" and the door opens, and then he says, "You could say that I have a magnetic personality!"
Data, tapping out and singing the "Life Forms" song on the console on the bridge... "...where are you?!"...Doo Doo Doo Doo Doo Doo Do!
@ArmoredSaint: I keep hearing Jack Black's name suggested to play Mudd in an Abramsverse film. I'm not sure that's a good idea. I don't think he could act a scene like that!
Jack Black as Mudd? Mike Myers has indeed the appropriate style. They would only have to mix is make-up of Inglorious Bastard with Fat Bastard's. Of course, it would make a Mudd who drops a lot of puns about Vulcan hears or Klingon foreheads.
I love this too, may favorite part is: Mudd: They've got no sense of humor, they arrested me! McCoy: Oh, I find that shocking. Mudd: Worse than that, you know what the penalty for fraud is on Deneb V? Spock: The guilty party has his choice, death by electrocution, death by gas, death by phaser, death by hanging. Of course Spock knows that!
That scene is comedy gold. Kirk is fumbling for words, trying to ad-lib an explanation for Spock's unusual appearance, while all the cop cares about is that he just caught two thieves red-handed. Hell, he's probably encountered weirder-looking characters than Spock in the line of duty!
One if my favourites is in DS9 when Worf discovers Keiko is pregnant again, the look on his face is priceless.