Deliberate mistake: When Adam kills himself and Hector at the end, the explosion sends a lot of water and parts of Hector into the air. But there is no sign of Adam's blood, body parts or clothing, despite the fact that the bomb was strapped to him. (01:16:10)
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.