Continuity mistake: When the satellite hits the spaceship, the shots from inside show the spaceship turning around clockwise, yet in the outside shots it is moving randomly: up and down, clockwise and anti clockwise.
Other mistake: The subway carriage stops by the Statue of Liberty ad, however, the one reflected on the cabin's glass is a VW ad.
Revealing mistake: Due to the blue-screen effect, part of Superman's glasses disappear when he is free falling with Lois.
Visible crew/equipment: Boom mic reflected on the hood of the cop car's, during the city attack.
Plot hole: Nuclear Man sends laser beams pointing his arms and destroys everything around, yet when inside the Daily Planet, he sends beams to the elevator but absolutely nothing happens.
Visible crew/equipment: When Lenny lets the guards in you can see filming equipment reflected in the rear-view mirror of Lenny's car.
Continuity mistake: When Superman flies towards a mountain, the lava on the volcano behind him disappears.
Revealing mistake: Watch the cars and the grey truck during the LexTv scene. Whenever Luthor appears on screen the same cars are driving on the street.
Continuity mistake: In class, Jeremy has his body bent over his desk. When the angle changes he is sitting properly on the chair.
Continuity mistake: At the Great Wall of China, Nuclear Man points his hands forward and explodes the wall in front. As of then, there's no continuity between the position of his hands and the location of the places exploding.
Continuity mistake: When Lenny leads his car with the cops in, the car is facing a very short road ending in a small mount of rocks. The following wider angle shows a very wide open space with grass and a huge cliff.
Visible crew/equipment: Nuclear Man descends close to the statue of liberty to fight Superman. For a brief glimpse his supporting wires are seen.
Revealing mistake: It is way too obvious that when Luthor holds the dollar bill, his arm and hand have been replaced by a robotic prop. His arm manages to move without involving all the bones connected to the shoulder and the back.
Revealing mistake: While Superman carries the Statue, there's a close shot of Nuclear Man showing his fingernails. The movie behind is playing the wrong way, and gives the sensation that Nuclear Man is flying backwards.
Audio problem: When Superman is wounded on the neck, and when the italian villagers are saved, the same male falsetto voice is used to add a (crappy and out-of-sync) "Oooh!" and "Aaaah!".
Continuity mistake: When Superman enters the UN hall, a back shot shows the people in the upper bleacher clapping, but a frame later, the front shot shows them sitting still.
Revealing mistake: The close shot of people panicking when the Statue is about to crash against the pavement shows an old woman with a red wool hat and a white scarf. Many scenes later, when Nuclear Man lands on Metropolis, the same lady is seen, wearing the same clothes.
Continuity mistake: While the crowd follows Superman to the UN, the man with the balloons is first seen far away from him, but half a second later he is very close.
Continuity mistake: When Nuclear Man is destroying Metropolis his hair suddenly changes from messy to perfectly brushed.
Continuity mistake: Seen from inside the train-driver's seat, the tunnel is darker and with lights on both sides, but a shot of the tunnel alone shows it brighter and with no lights on the left side. Then it cuts back to the previous angle and everything changes back.
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.