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.
Award Snub: James Doohan said he thought Ricardo Montalban should have received an Academy Award nomination for Best Supporting Actor, complaining that movies like Star Trek never receive nominations for major awards.
Foe Yay: There is some palpable sexual tension between Kirk and Khan, even if it's just rivalry. When Khan is preparing to fire on the Enterprise, he seems to be close to ejaculating at the prospect.
Franchise Original Sin: After this movie, nearly every Star Trek film would end up being built around one particular villain (the major exceptions being IV, which doesn't have a villain, VI, in which General Chang is just the most visible member of a large conspiracy, and Star Trek Into Darkness, in which Khan and Admiral Marcus are both villains on opposing sides) for better or worse. The failure of Star Trek: Nemesis had shown how worn-out the formula had become.
Fridge Brilliance: In the Director's Cut, Midshipman Preston claims that if Kirk can't see that the Enterprise is the finest ship in the fleet, then he's as blind as a Tiberian bat. Did the Plucky Middie just make a pun on Admiral James Tiberius Kirk's middle name to his face? No wonder Kirk was impressed. Kid's got balls the size of Kirk's ego.
Fridge Logic: Khan seems to believe that he's buried Kirk alive, and that he'll be stuck in that asteroid forever. It seems likely another Federation starship will be along well before anyone gets really uncomfortable.
The implication is that, now that he's dealt with Kirk, Khan is going to go make a power play for control of the Federation using the Genesis device (no doubt replicated through superior Augment intellect). Nobody is going to have time to come looking for Kirk. Since the Enterprise is the only ship in the vicinity, and Khan believes her to be helpless, he likely figures he has time to waltz over and finish her off; he can then take his time to loot the station of the resources he needs to replicate the torpedo and sustain the Reliant, torturing Kirk on the side should he wish to do so, and by the time Starfleet could get another ship out there, Khan could be long gone, plotting his next strategy in safety.
He Really Can Act: "KHAAAAAN" aside, Kirk has a lot of quiet, subtle character moments, and William Shatner really delivers. Nicholas Meyer took credit for this one, forcing re-take after re-take until Shatner finally got tired of "performing" and actually did some acting.
It probably should be pointed out that with even the "KHHHAANNN!" line, the character of Kirk was likely deliberately overacting to fool Khan.
But it's also true that he really was angry and frustrated that Khan would just keep on going killing and most likely intended to go on to finish off the Enterprise. So if anything, he is at least letting out his aggression.
Yes, he was pissed, and he vented a lot of anger, but he'd made it pretty clear to Khan that the Enterprise was in no position to rescue the landing party anytime soon, so he had to make sure Khan was convinced of their helplessness.
Also true of Ricardo Montalban, oddly enough. At first glance that may seem absurd. Montalban was a movie star in The Fifties and also starred in a long-running hit TV series, Fantasy Island, that was even still on the air when Khan came out. But most of his leading man roles from the Fifties were unchallenging Latin Lover stereotypes, which Montalban hated. (One of these films was even named Latin Lovers!) In The Sixties, Montalban switched to doing guest star roles in television (like Star Trek: The Original Series) which is great for expanding range but less so in gaining/maintaining name recognition (few people would be watching an episode of a TV series mainly because Montalban was guest starring in it). As for Fantasy Island, it was always more of a popular hit than a critical one, with Montalban admitting his character there was mostly Mr. Exposition. When Khan came out, Pauline Kael (who was one of the toughest, harshest film critics of the time) was so impressed by his performance that in addition to her (mostly) favorable review she wrote up an admiring profile on Montalban. Nicholas Meyer would later say that Khan showed Montalban was capable of playing King Lear.
Heartwarming In Hindsight: Spock says that Kirk's solution to the Kobayashi Maru "was, shall we say, unique". In Star Trek, when Kirk says to Spock Prime that what he's doing, disclosing future history, is cheating, Spock answers that it's "A trick I learned from an old friend."
Ho Yay: Kirk/Spock (surprise, surprise). Notably at the end of the movie: Kirk and Spock press hands against the glass as Spock dies. Spock's death has a huge impact on Kirk (which continues into the third movie.) Kirk says he's never truly faced death before, "not like this," even though his brother died during the series. Kirk would go on to say that he had lost "the noblest half of [him]self" - and look closely. Kirk and Spock are Vulcan kissing through the damn glass!
"When I try to explain slash to non-fans, I often reference that moment in Star Trek: The Wrath of Khan where Spock is dying and Kirk stands there, a wall of glass separating the two longtime buddies. Both of them are reaching out towards each other, their hands pressed hard against the glass, trying to establish physical contact. [...]And, I tell my nonfan listeners, slash is what happens when you take away the glass."
Khan's obsession with Kirk can be interpreted as Foe Yay.
Nicholas Meyer: Kirk did not lie awake thinking about Khan; Khan lay awake thinking about Kirk.
It takes three men to restrain Kirk from opening the door holding back lethal radiation just so he can be with Spock in his final moments. And what's more, Bones and Scotty knew full well they would have to do it and were already prepared when he arrived.
Moral Event Horizon: Well, by the time he shows up, we know Khan Noonien Singh is a bastard, if a magnificent one, especially if you've seen Star Trek: The Original Series. But when he starts putting worms in people's ears, you realize that perhaps he's kicked it up a notch on the Evulz scale; but you say to yourself that he's intelligent and charming and surely he can be reasoned with. But...when he listens to a minion kill himself and doesn't so much as blink, merely ordering the surviving minion to hurry up and carry out his orders, you realize this guy ain't coming back from the horizon. Khan also certainly reminded the audience that he was evil when he activated the Genesis device instead of surrendering to the Enterprise. He knew that he would take down at least one of the Enterprise's own, if not the entire freaking ship. He would have wiped out many people just to satisfy a longtime desire for revenge. It wound up that Spock, Kirk's closest friend, sacrificed himself to save the Enterprise.
Newer Than They Think: "The needs of the many outweigh the needs of the few, or the one." It sounds like an old Vulcan proverb, but more reliable sources point to this movie as the origin of the phrase.
On the other hand it sums up the philosophy known as utiliarianism, which is about two hundred years older than the movie.
Arthur C. Clarke considered listing Khan as one of the best science fiction movies of all time. Considering Clarke's far greater enthusiasm for 'hard' science fiction, any recommendation at all suggests a pretty strong film.