When the second fire engine crashes, the model turns on its side and very obviously falls off the table on which it was being filmed. There is nothing depicted on screen from which it could have fallen off.
The buildings attacked by Godzilla in the model shots, from the way they "slide" on their bases and explode upwards in one whole piece, do not appear to have any foundations (or, indeed, to be fixed to the ground in any way at all).
As the monster lumbers through Tokyo, the emergency officials are shown evacuating their office just before the building is wrecked by Gojira. After the office has been destroyed, a shot shows the monster's tail outside the wrecked building, and the wires holding up the tail are plainly visible.
At the meeting of officials after the scientific party's return from Odo Island, Dr. Yamane shows what he calls a photograph of Gojira. First, though it's supposed to have been taken by the party while on the path up the mountain, from its angle the photo could not have been taken by anyone standing on the island, only from the air. Second, it's obviously a painting, not a photo. Third, the face and head of the monster in the "photo" do not at all look like the monster's actual face or head.
The goof item below may give away important plot points.
When Serizawa demonstrates the oxygen destroyer to Emiko in his lab, he switches on a machine that lights up before the device is used. But when he uses it against Gojira in Tokyo Bay, he just releases the "destroyer" without having to use any equipment to make the device work.