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.
Continuity mistake: When Nuclear Man arrives on Earth, he enters a shady place which makes him loose his power and cools his body, producing some frozen-like smoke. The next two times that he enters a dark place (inside the elevator and in space) he doesn't produce the smoke. FX budget problems here?
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.