Revealing mistake: In the first confrontation between Superman and the Zod Squad, Superman gets chased out over the water. In the first two shots in this chase, if you look at the bottom right-hand corner of the screen (at least in the letterboxed version) you can see the wake of the boat that they used to make these two pull-back shots. (01:34:40)
Factual error: The moon has no atmosphere. Yet, the Ursa and Zod are able to speak to the astronauts. We even hear wind as Zod kicks one of the astronauts in the air and snaps his cord. We also hear the explosions when the lunar lander is destroyed. If the moon has no atmosphere, then the super villains can't speak and there should have been no sound in this part of the movie.
Other mistake: When Ursa, Non and Zod first arrive on earth, watch out for the horrible lipstick marks on Ursa's teeth as they walk away from the lake.
Deliberate mistake: When the villains rampage the White House, the flag's base remains on the ceiling, and the flag pole falls inside the house. So the final scene with Superman carrying the flag and its base as if it had been previously stolen or destroyed, is nonsensical, but it looks very patriotic indeed.