Continuity mistake: When Nuclear Man steals the Statue of Liberty, its pedestal looks normal. When Superman puts it back in place, the pedestal is missing a huge chunk (surely because of some FX going wrong). In further shots, it's fixed.
Revealing mistake: When Superman and Nuclear Man fall down the building, their capes remain stuck to their backs, defying the laws of gravity, and thus revealing that the scene was shot with them standing up and then turning the camera around.
Continuity mistake: When Moscow's Red Square scene begins, the tanks and soldiers parade in front of the authorities (a combination of matte painting and a real set). When Superman arrives, a second tree has appeared next to the one on the left. On the right side, an unfinished painting has replaced the previous huge wall, trees, and decor. (International theatrical, TV, and LaserDisc releases).
Visible crew/equipment: When Superman saves the day in Moscow, the edge of the set and the backlot behind can be seen on the sides of the wall where the politicians stand. One can even see the set's scaffolding on the right. (International theatrical, TV and LaserDisc releases.)
Factual error: If the Russian cosmonauts need special suits to survive out in space, so should Lacy when she is abducted by Nuclear Man. (00:02:50 - 01:20:20)
Continuity mistake: Nuclear Man dents the floor with his footsteps, but in a wider angle the floor is intact.
Continuity mistake: Two firemen are seen away from their truck, but when the angle changes six firemen have appeared seemingly from nowhere.
Continuity mistake: When Superman is walking with the kid towards the UN, they pass by a fire hydrant and one can see the street ahead is empty. A shot later another hydrant suddenly appears but the white benches disappear.
Suggested correction: The fire hydrant in question is actually two. The one they first pass is slightly faded with chipped paint. The next shot you see a different one that is fully painted red. The next shot shows the crowd passing the first hydrant seen.
Continuity mistake: When in the subway, Superman is standing by a 6 foot poster of the Statue of Liberty. When the shot changes to a flying Superman, he is a third of his size and the poster is huge.
Continuity mistake: Clark enters the barn to fiddle with his green crystal. When he exits, a huge amount of stuff from his childhood has magically appeared around the place.
Continuity mistake: Before lava covers the Italian village, furniture under the two arches on the right, as well as a vase, cart wheel and table on the left disappear between shots.
Other mistake: The same front shot of Superman flying is used over and over and over again.
Continuity mistake: While the lava flows on the Italian village, a wooden cart with hay on the left suddenly becomes a red car.
Visible crew/equipment: In the quarry, when the cops' car flies up the air you can see the ramp used to lift it on the lower left side of the screen.
Other mistake: When Clark Kent enters the subway, the set is the same as the rampage later in the film; only that the second time we see this set, the subway entrance has vanished.
Continuity mistake: When Superman and Nuclear Man fight on the balcony the high powered binoculars disappear.
Revealing mistake: In this scene, all the SWAT cops have the same type of rifle, except for the guy that gets his melted by Nuclear Man, which is a thick rubbery prop.
Other mistake: Nuclear Man is shown blowing up the Great wall twice, but it's the same shot reversed.
Continuity mistake: During the volcano eruption the animal on the right (goat or sheep, I don't know) keeps changing from brown colour to white.
Continuity mistake: The cab's hood, crumpled by Nuclear Man's power beam, magically smoothes itself a frame later.