Continuity mistake: In the beginning of the UN sequence, we see a brief outside shot of the real UN building, but the further shots of the fake building have none of the elements from the shot we just saw: Flags, white stone wall.
Revealing mistake: While the subway train is gone wild, there's a brief shot of bystanders on the platform. Then the train keeps going on for several seconds, and when it finally comes to a stop the actors are the same ones used previously.
Revealing mistake: When Superman drops the elevator containing the imprisoned Nuclear Man on the moon, he flies off and lands to straighten the American flag. In this shot you can plainly see the wires helping him land.
Continuity mistake: When Superman is entering the UN, an Arab man greets him and turns his head around from right to left. In the following closer angle he repeats the movement.
Continuity mistake: Superman is on the moon, about to cause the sun eclipse, and flies to the left, as if he were to exit the moon. In the next shot Superman is seen coming from the void towards the moon again.
Continuity mistake: When the Soviets are about to launch the missile in the snowy field, there are two soldiers on the right and a third with a black hat running away. A second later there are three soldiers plus the one with the black hat who's now there.
Continuity mistake: When Lois is packing Superman's cape and Lacy says "I had nothing to do with it", there's no continuity with the passers-by seen behind. A kid in a red t-shirt, for example, appears from nowhere.
Continuity mistake: When Superman saves the girl from the Great Wall of China and starts rising, a woman with a white hat behind lifts her arm to say goodbye. In the following angle she is repeating the movement.