Awesome Music: As with the rest of this film, the music is...hotly debated, with one exception. This track (featuring a spoiler title) is widely considered really fun. When it or either of its two variants gets going, you know there's one hell of a scene approaching.
Considered the best by some for its cast of kaiju and nonstop action. Considered the worst by some for its focus on the humans instead of the kaiju and the action instead of the story. It doesn't help that many of the scenes with the human characters are taken wholecloth from other famous films such The Matrix, Star Wars, Independence Day and X-Men, sometimes frame by frame. One notable example is the character of Shinichi Ozaki who strongly resembles actor Keanu Reeves and at one point in the film stops laser blasts in mid-air in a manner very similar to Neo's most famous scene in The Matrix Reloaded.
Monster X being a disguise for Keizer Ghidorah. Some view it as a case of They Wasted a Perfectly Good Plot in that Godzilla didn't actually get to fight a new monster in the movie, while others view a showdown between Godzilla and his Arch-Enemy as a fitting send-off for the franchise at the time.
As alluded to above, the music. Was Keith Emerson out of place to score a Godzilla movie, or does his approach add to the fun campiness of the film?
Gigan's Millenium design. Some fans love it for being more threatening, sleek and overall futuristic-looking than the Showa version, while others think this redesign pretty much got rid of nearly everything that made Showa Gigan so fondly remembered.
The alien enemy X is a Large Ham who has a Villainous Breakdown after each of his monsters die, but that doesn't hide just how evil he is. His boss, The Controller, wanted to take Earth peacefully and was simply a Well-Intentioned Extremist trying to save his people. X offs him the moment he gets the chance and takes over, instantly letting loose his army of monsters and begins the destruction of human civilization. While many alien invaders in Godzilla movies have done just that, it's the sadistic glee present in X as he watches the horror he's unleashed that makes it clear that, unlike his more peaceful predecessor, he's a sadist who enjoys what he's doing! When he captures the Gotengo at the climax of the film, he orders his men to slaughter the crew except the ones he wanted personally. Where his predecessor took prisoners even when he could've simply let the victim die, X has absolutely no regard for life at all. During the Final Battle of the film, he takes his time to enjoy the Curbstomp Battle he's inflicting on his opponent.
X's own (literal) Dragon, Monster X, initially seemed to be just another weapon in the would-be conqueror's arsenal, dutifully battling Godzilla and Mothra on behalf of his demented master. Throughout the fight, however, the creature would display a degree of independence and cruelty unknown in X's other slaves, pinning Godzilla down so that he could be slowly cut to pieces by Gigan's chainsaws, and seeming to enjoy the battle. It was only after X and Gigan's deaths, and the revelation that Monster X had not been mind controlled, however, that the being's true evil shone through. Shapeshifting into its real form of Kaiser Ghidorah, the alien dragon continued to try and kill Godzilla, using Vampiric Draining to slowly suck the life out of Earth's protector. Only too happy to try and destroy the world, and of his own free will, Kaiser Ghidorah proved himself every bit as evil as the Ghidorahs who came before him.
Gigan finally making a reappearance with a far more menacing design was well-received.
For human characters, Captain Gordon has been very well-received. Being played by Don Frye helps.
Inferred Holocaust: The heroes defeat the Xillians and Godzilla departs peacefully but pretty much every city on the planet has been devastated and millions if not billions have been killed as well as the fact that several monsters are still alive.
Take That, Scrappy!: There's a reason that Zilla is the first kaiju to get a complete smackdown, and it's to establish that the character doesn't hold a candle to the "real" Godzilla.
They Changed It, Now It Sucks!: Some fans aren't exactly fond of Gigan's Millenium design, even calling it way too overdesigned and gritty for their taste. It doesn't help that this design puts more emphasis on Gigan's robotic qualities rather than his biological ones, which makes him look somewhat less like a cyborg for some.