My favorite lines (the first five examples are notable for how they turn what seems like a cliche on its head to mean something utterly different):
"I do not know." Spock's expression here is memorable, after the Kobayashi Maru. I find myself repeating that intonation in everyday life - a combination of hostility, suspicion, anger, and calculation among other things.
"Live long and prosper" by Spock to the Vulcan Science council. Obviously a famous and memorable line, in that context.
"Fascinating." Again, it's the situation that makes this phrase so much more meaningful than usual: it's the same words, but a totally different meaning than usual (like "I do not know" and "live long and prosper").
"Is there a problem, officer?"
"At ease".
"Minutes."
"Hello, Christopher. I'm Nero."
"Refusal would be unwise."
Not really a a great line, but memorable for other reasons is "Captain, we have visual" (i.e. the story behind it).
"Romulans" (Kirk to Uhura in the corridor with McCoy).
"Numb tongue" (same scene, and the conclusion of course).
"Wiktor wiktor"
"I presume you have prepared new insults for today"
