When Superman crashes down to Earth after killing Doomsday he has a tear in the shoulder of his costume, after he dies in Lois' arms the tear is gone.
When Jimmy is looking at the newspaper in the alley with Lois, there is a small trashcan behind him. It's an olive green color and is dented. Then after he throws the paper at Lois and begins to walk off, the can is now gray, larger, and no longer dented.
Doomsday has a bone spike protruding up from each of his 4 knuckles. But when Doomsday grabs Superman by the head and draws back to punch him hard in the face, the spike on his thumb is missing. Superman catches his fist, and in that shot you can see two more of the spikes have vanished, and Doomsday only has one over his middle knuckle.
When Doomsday slams Superman down into the subway, the hole they make in the street is right in the middle of a crossing. But a minute later in the film as civilians are walking up to the hole, there are no lines anywhere around it. Moments later after Jimmy takes a picture of the hole, the shot cuts to a far angle and now the hole is next to the crosswalk lines, and in that shot the stairway entrance to the subway station also vanishes. The road also becomes much wider with more lanes, and the red truck turns around.
In Metropolis on the bridge, Doomsday grabs a soldier and breaks his neck and skull. In that shot, the soldier's arms are inside Doomsday's hand and his hair is a light brown or gray. But then when the shot changes as Doomsday begins to throw the soldier to the side, the soldier's arms are suddenly outside of Doomsday's hand, and his hair is now black.
Why don't the other hero's take over after Superman dies??? He is clearly dead, yet not one of the league members is seen helping out, and trying to keep the city safe.
What does the Justice League not appear in this film??? There is no explanation as to why this happens. Doomsday is one of most powerful characters in the DC Universe, and would certainly attract there attention. They don't even interfere, and just let superman die.
Lex Luther is seen killing someone. Luther often tries not to kill anyone at least directly. He may be a rich, snobbish guy but this contradicts his character.
When Lois is trying to seduce Lex Luther, he falls for her seduction. Yet, this would be highly stupid. One, Lex Luther and Lois Lane have never liked each other, and Luther should been more suspicious. Also, a lawsuit could have been filed against Luther, for sexual harassment and maybe worse. Luther tries to keep a boy scout image, and this would contradict that image. He is simply stupid, and has no positive history with Lois Lane. This just wouldn't happen.
Why doesn't Lex Luther have more security??? Lois simply walks in and sets down. Luther is a very rich man, and would have a lot of security, including guards and security cameras. Yet, no one stops her and she walks in like she owns the place.