Continuity mistake: When Superman is given a baseball bat, the angle changes and the position of the shadows reflected on the barn change completely.
Continuity mistake: When the Russian spaceship is spinning round, the bright and orange light inside and its minimal decoration are totally different to the inside shots, with a dim light and cables and stuff everywhere.
Continuity mistake: Before Superman throws the heavy weights at the gay gym instructor, two small weights are laying parallel on the floor. A second later one has moved underneath the bench.
Continuity mistake: When Superman flies towards the Russian cosmonaut his space suit changes from silver to green.
Continuity mistake: When Nuclear Man lets go the the Statue of Liberty, he disappears in the immediate wide angle shot.
Continuity mistake: When Superman is about to take the Russian cosmonaut back inside his spaceship, a first angle shows him very close to the doorway. A shot later he is farther away.
Continuity mistake: Before Luthor introduces Nuclear Man to Superman, they have a chat next to the bar. The bottles and figurine on the back changes positions between shots.
Continuity mistake: When Superman misses hitting the baseball, the ball's position on the floor changes between shots.
Revealing mistake: When Superman returns the cosmonaut back in the spaceship, Superman's head becomes transparent, and the cosmonaut's helmet becomes wobbly, all because of the use of blue-screen.
Continuity mistake: After Superman prevents the Statue of Liberty from crashing against the street, when he passes Central Park, there's a wide angle of the Statue flying and Superman is nowhere to be seen.
Continuity mistake: When Superman arrives at Lois's balcony, a plant on the left is away from the door. When Superman leaves and Lois asks him to take her with him, the plant is next to the door.
Continuity mistake: When Lacy is seducing Clark in the office, the things on the table change positions between shots. Watch how the glasses move around, for example.
Continuity mistake: Nuclear Man is destroying Metropolis and always, behind him, there is a white truck and a lot of cars. But in one of the shots the cars have vanished and a lot of people appear.
Continuity mistake: After Superman straightens the Russian spaceship, he flies away to save the cosmonaut. In the following wide angle he is flying towards a different direction.
Plot hole: After Nuclear Man slides down the chimney of the Nuclear Plant the lid of the reactor is open and then automatically closes. This whole premise is totally absurd, not only for security reasons, but for logical ones too.
Visible crew/equipment: Lighting screens are reflected on a hotdog seller's glasses when Superman walks to the UN.
Revealing mistake: Many shots of Nuclear Man flying are reused throughout the movie.
Continuity mistake: Lenny tells the prison guards they can check his sound system, and the guard on the back grabs the shotgun with his left hand. Shot changes, and the shotgun is back in his right hand, and he is repeating all previous movements.
Continuity mistake: Luthor and Lenny drive past a red and white truck and a Maglite billboard. A shot later, they drive past them again.
Revealing mistake: When Clark receives Lex's signal, he rushes out of the room where Lois and Lacy are. First, he crouches behind a couch and disappears. Then the camera pans to the right, and now he's Superman. Problem is, one can still see Clark's stand-in crouching behind the couch, revealing how the trick was achieved.
Answer: It was made on a very low budget. Golen - Golbus productions bought the rights to Superman. They were mostly known for B-Movies with not so big name stars. It was there attempt to play with the big studios. Plus at least 45 minutes of scenes were cut out, with major subplots.
In addition to budget cuts, they kept shortening the runtime, meaning scenes needed to be cut. The comic book adaptation has the uncut scenes and makes much more sense.