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.
Better on DVD: When you watch the film again, you can see just when Spock slaps the patch on Kirk - and see the patch, too. The Klingons weren't very thorough, huh?
Always bugged me. You're telling me the Klingon Empire are supposed to be the villains of the story, but they don't so much as take their prisoners' clothes away? Heck, they even give them extra clothes on Rura Penthe, though, granted, they'd probably die of hypothermia if they didn't. However, they still get to keep their Starfleet Uniforms...why?
Harsher in Hindsight: The film was written at the latest early in 1991 (to give time for filming and post-production for the December release date). The film was essentially about a dual coup attempt against both the Klingons and the Federation so that Blood Knight elements in both could continue their war. In August 1991, there was a coup attempt against Gorbachev in the USSR, by Blood Knight elements in the CPSU who wanted to continue the Cold War.
Ho Yay: Kirk and Spock are in the same film, of course there is. There's one particular scene in a corridor that was a deep breath away from being a kiss.
Life Imitates Art: The plot of the film, whereby hardliners attempt a coup against the moderate leader in the hostile empire, was nearly repeated as the film was being edited: in form of the August coup against Mikhail Gorbachev, on whom Gorkon was based, by hardliners who wanted to go back to the good old Cold War days.
Memetic Mutation: "Only Nixon could go to China" gets used frequently in political discussions.
This is Older Than They Think. The phrase has been in use since at least 1971, before Nixon actually went to China.
Nightmare Fuel: Spock's reaction to Valeris' betrayal. It's chilling to think of how furious he must be to display that much emotion.
One-Scene Wonder: Christian Slater has a cameo as the crewman who wakes Sulu up in the middle of the night. The scene was originally written for Grace Lee Whitney as Janice Rand; Slater only got the part as a favor to his mother, the casting director. Also a case of Promoted Fanboy.
When Chang's Lieutenant or Weapons Officer presses the button to fire torpedoes (installed in a box attached to a wall) you can clearly see the entire box shift position as if poorly glued down.
When Chang is quoting Shakespeare as he watches the Enterprise fly past, the Enterprise looks like what it is: a model, right down to the scored detail lines and paint job. Especially jarring when compared to the other exterior shots of the ship in the movie.
Not all the exterior shots, in fact not many of them at all. This film's visuals of the Enterprise are completely different in style altogether to the other movies, and suffer by comparison. Despite using the same filming model. Especially rubbish is the "warp to camera shot" after Kirk says "Come on, I need you" to Spock.
There was also no background in the Bird-of-Prey's view of the Enterprise, either, just a completely black space devoid of stars.
The Klingon blood is depicted with early 1990s CGI in all of its imperfect glory. To be fair, it's been twenty years and the technique has yet to be perfected.
It's interesting to compare the blood effects in this film with "Barge of the Dead", an episode of Star Trek: Voyager which aired only 8 years later, to see how much CGI blood (and indeed, CGI liquids in general) had been improved.
The Zero-G blood drops were also one of the first uses of an at-the-time new CGI technology called "metaballs" - a new way to make smooth, blobby, and more organic objects. As much a tech demo proof-of-concept for the technology as a special effect.
The Untwist: General Chang was behind the attack on Gorkon's ship. Given his attitude when he's first seen coupled with the fact that he had the recording of Kirk's statement he would never forgive Klingons, it's not much of a surprise.