These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Just for a moment, let's look at the movie from Kruge's point of view. His talk about Kirk being "an enemy of intergalactic peace!", and his fears that "The Federation, in creating an ultimate weapon, has become a gang of intergalactic criminals!", both sound like loony rants. But what if he believes that? What if Kruge is doing the wrong things for the right reasons? If he is under the mistaken belief that the top secret Project Genesis is really a super-weapon, then everything he does could be interpreted as a loyal subject of the empire trying to protect his people from their sworn enemy. Remember that the Klingon/Federation conflict was always intended to be a allegory of Cold War tensions, and that this movie came out at arguably the height of those tensions.....
Was there a more in-series triumphant moment than immediately after "We have cleared space doors"?
Triumphant, yet chilling, knowing what fate lies ahead for the Enterprise.
"Funny Aneurysm" Moment: On the Special Edition DVD release of Star Trek III: The Search for Spock, in the text commentary by Mike Okuda for the scene where the Starfleet commander tells Kirk the Enterprise is to be decommissioned because she's twenty years old, he remarks that NASA has less trouble with old spacecraft, as the Space Shuttle Columbia was still flying despite being over twenty years old. Shortly after the DVD's release, the Columbia burned up on re-entry, killing all on board. To make things worse, the shot of the Enterprise burning up in the atmosphere resembles the Columbia disaster.
Hell Is That Noise: The buzzing sound that ensues when the Klingons jam the Grissom's communications.
Ho Yay: Kirk had always put his ship before everything else in his life - until it came down to a choice between his Silver Lady and his First Officer. He chose Spock. And that's not gay... how?
How about the whole conversation between Kirk and Sarek at the beginning of the movie, which implied that Kirk and Spock were once romantically involved.
Search For Spock holds the trophy for the most Ho Yay of all the films. How could it not with such gems as:
Kirk and Superior Officer: "But if there's even a chance that Spock has an eternal soul... then it's my responsibility." "Yours?" "As surely as if it were my very own."
Kirk (to Sarek): "Your son meant more to me than you can know."
Kirk and Sarek: "What I've done, I had to do." "But at what cost? Your ship. Your son." "If I hadn't tried, the cost would have been my soul."
Spock: "Jim. Your name... is Jim." (It works in context: the only thing Spock remembers is Kirk. Daww.)
Bones' scene with Spock's body. "I don't know if I could stand to lose you again" indeed...
And then there's Hikaru "Don't call me 'tiny'" Sulu. Oh my!
Let us not forget the scene where Kruge shows his underlings the Genesis presentation, followed by him whispering his plans in the ear of Maltz with an intensity which sounds like he's going to end it with "...and then you and I will have the most epic angry sex in the history of the galaxy."
Kruge's line about Kirk being "Enemies of Galactic Peace!" is pretty corny when you think about it. If the movie was more popular then I'm sure more people would pick up on this beaut'.
"The Federation in creating an ultimate weapon has become a gang of intergalactic criminals!"
Klingon bastards, you killed my son!!!! (though most of the audience sympathized)
Narm Charm: Kruge, with Christopher Lloyd's rambling Doc Brown voice, especially to the nostalgic viewers who originally watched the film as kids — you can't help waiting for him to exclaim "Great Scott!" after every one of his lines.
Dom Irrera's standup included viewer's confusing him with Reverend Jim.
Dom:(as Lloyd) So, you want to take the Genesis device? ... Hokey doke!
What an Idiot: Scotty tells Kirk that the automation is burning out as he did not anticipate combat. Enlightened society or not, the Federation would likely not be too kind to seeing its military hardware getting stolen, leaving the very real possibility that they'd need to defend themselves.
What The Hell, Casting Agency?: Christopher Lloyd as the balls-out ruthless Kligon Commander Kruge. He was prior and after that mostly associated with comedies and light dramas. But hey, he did pretty damn well playing the most brutal Klingon captain ever seen in the Star Trek franchise. Every other Klingon captain either had minimal screentime or wound up earning at least some sympathy. Kruge was a monster, and every kick he took to the face from Kirk's boot was pure audience satisfaction. He is also the only Klingon to call Kirk on his bluff when Kirk demands his surrender.
Kruge: He's hiding something. I must have dealt him a more serious blow than I thought.
And parts of the blown-up Enterprise 1701 models were used in the aftermath of the Battle of Wolf 359 sequence in "The Best of Both Worlds", because they were so detailed they could be shown very close to the camera without looking fake.