Revealing mistake: When NuclearMan goes to destroy the Great Wall of China, there is a brief glimpse of him wearing support wires.
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.
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?
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.