Continuity: Godzilla noticeably changes size throughout the movie. For example, he is shown lairing in a subway tunnel, but he cannot fit into the Hyde Park tunnel (taxicab scene), which is significantly larger than a subway tunnel. Also, compare the size of Godzilla's eye next to Matthew Broderick during the fish scene and Godzilla's death scene. The eye is way bigger in the latter.

Continuity: When the Mayor of New York is talking to the crowds and people can feel the ground bouncing, Godzilla is a fair bit away, but when he is stomping through the streets the people don't feel it at all - shouldn't they be thrown into the air from that proximity?

Continuity: Near the end of the movie there are 3 USAF planes.  The lead plane shows a graphic of 4 missiles, 2 of which he launches at Madison Square Garden. Then later, and without landing, the 3 planes fire at Godzilla.  2 missiles from each plane on both passes.   This means the lead plane fired 6 out of the 4 missiles.

Continuity: When Godzilla is chasing the taxi, the taxi crashes through a chainlink fence. As it rounds the corner shortly after going through the fence, the sign on top of the cab comes loose and is being held loosely by a wire toward the front of the cab. You can clearly see that the sign is loose, but a few shots later, the sign is attached firmly in place atop the cab.