Plot hole: Superman traps the supervillain (whose power depends on sunlight) inside of an elevator to incapacitate him. Superman then ripped the elevator out of the building. He then plants it on the far side of the moon. Later on, sunlight starts to shine into the elevator through a slit at where the doors meet. The villain of course recharges and comes after Superman again. Now, if light could get through that crack there, then why couldn't it get through when the elevator was ripped out of the building in BROAD DAYLIGHT?
Factual error: When the villain takes the reporter into space (the earth is just a circle off in the distance), then weakens and lets go of her, not only does she start to fall, she starts to fall straight down and has to hang onto him. (Oddly enough, HE's NOT falling). Did somebody inject normal earth gravity into mid-space?
Plot hole: When Nuclear Man sees a picture of Lacy Warfield on the newspaper he decides to go after her. When confronted by Superman he asks "Where is the woman?" Superman responds that he'll never find her. How could Superman know which woman Nuclear Man was talking about? There was no indication that Superman knew that Nuclear Man was looking for Lacy.
Revealing mistake: When Superman saves the people that Nuclear Man throws up into the air, as they lower you can see the support wires holding them up.
Visible crew/equipment: When Superman and the super villain are fighting on the moon, when Superman flies in and lands at the beginning of the fight, you can see the cables that are suspending him.
Revealing mistake: When the net full of nuclear missiles is thrown into the sun, the footage of the explosion is exactly the same as the explosion of the single rocket with the DNA of Nuclear Man attached.
Continuity mistake: When Nuclear Man arrives to earth, full of energy from the sun, his footsteps don't glow; neither do they when he arrives to the balcony, several scenes later. Then a couple of seconds after they glow but stop glowing when he walks towards Superman and keep that way for the rest of the film.
Continuity mistake: When Nuclear Man is destroying Metropolis, the same stuntman with a gabardine and a hat is seen close to a shop window, then a second later in the middle of the street and a second afterwards getting thrown through the air with a group of people. He looks and dress just like Clark Kent, guess it's his stand in doing some extra work.