Awesome Music: Unlike Superman II and Superman III, this has a substantial amount of original music; John Williams didn't score this one - his longtime friend and regular orchestrator Alexander Courage did - but he did write new themes for Jeremy, Lacy Warfield and Nuclear Man (and say what you will about the guy, but Nuclear Man does have a pretty kickass leitmotif).
Broken Aesop: Superman's stone-serious lecture about the benefits of public transportation falls a bit flat when he gives it after stopping a deadly, out-of-control metro car.
Critical Research Failure: While overlooked at the time, modern audiences, as a result of countless crime dramas, are aware that only the root of the hair contains DNA, and thus Lex would get nothing of value from his theft of Superman's hair, as it can be clearly seen that he doesn't grab the root. Then again, maybe kryptionian hair is different.
A particularly horrifying example, where Clark Kent goes to a gym with Lacy Warfield and pretends to injure his back lifting weights. It's not a direct parallel, but the image of Christopher Reeve holding his spine and wincing in pain is very eerie.
At the end of the film, Lenny gets dropped off at a boy's home, with a priest saying "We take care of boys." Flashforward to the child abuse scandal in the Catholic Church, and that delivery becomes really creepy.
Sequelitis: The fourth and last film in the original Superman film series, this is the one with the most hard knocks/contempt and the only one to fail at the box office. Minus the Box Office Bomb part, history would repeat itself with Batman & Robin falling into this same category.
Superman IV is arguably funnier than Superman III; it has Narm in spades, and Unintentional Comedy > Failed Gags every time.
The mere presence of Gene Hackman can make anything a little better, and his Ham and Cheese here, as both clever Lex Luthor and the voice of all-brawn, no-brains Nuclear Man, is quite tasty. Jon Cryer's Totally Radical Lenny Luthor is more bearable than it otherwise would have been with Hackman to play off of.
A human woman breathing perfectly finein space, as it was originally intended for her to be in the skies above Metropolis.
Superman "rebuilding" the Great Wall of China with his... vision?
Just about every shot of Superman flying is the exact same shot.
The film actually rewinds to simulate Nuclear Man blowing the NYPD cops backwards with his laser vision.
Most of the nuclear missiles shown onscreen are painfully obvious props.
Needless to say, the film's visual effects landed the department a well-deserved Razzie nomination that year.
Surprisingly Improved Sequel: As horribly received as the film was on its initial release, many have come to regard this as a better sequel than Superman III in more recent years, due to the fact that this film at least took itself relatively seriously and tried to have a genuine (if hamfisted) message rather than the buffoonery that made up most of the previous sequel. Few people actually claim it to be good, mind, but it does get a lot of slack that III doesn't.
Superman and Nuclear Man's capes flap in the "wind" in space, and, as mentioned in Special Effects Failure, a human woman breathing perfectly fine in space!
The movie itself is considered one big They Just Didn't Care moment by many. The special effects alone are a travesty for the world's most iconic superhero.
Gene Hackman hated wearing a bald cap in the previous movies, so it was established that Lex Luthor normally wears a wig. Even though Lex is presumably still wearing one in this film, no effort was made to cover up Hackman's actual bald spots (it's safe to fault the makeup artists for this one, since Hackman himself definitely didn't care by this point).
Took the Bad Film Seriously: Christopher Reeve had to have known that the Superman film franchise was on its last legs when he signed up (with stipulations) for Superman IV: The Quest For Peace. To note, Reeve would only take the film if several conditions were met, one of them being a strict anti-nuclear message. While other members of the cast understand how bad the script is (Gene Hackman was there for a check and Jon Cryer was camping it up), Reeve gives it his all and delivers the only emotionally honest performance in the film, which is especially evident in the scenes where he prepares to sell the Kent family farm, and the sequence where he delivers a stirring speech to the United Nations. It's enough to make the viewer wish that the film wasn't screwed over with the most ridiculous villain ever seen in a comic book film (Nuclear Man).
Too Dumb to Live: As shocking as the Statue of Liberty being dropped on you might be, good citizens of Metropolis, if you have enough time to point up at it and yell "Wouldja look at that!", you probably have enough time to try running away from it, too.
What an Idiot: Apparently, no one in Metropolis bothered to install dead-man switches into their subway trains.