When Mothra and Godzilla are fighting, in one shot a smokestack breaks and falls over. In a wide shot, the smokestack is unbroken.
When the Mothra larva grabs onto Godzilla's tail and Godzilla flings it around, the skin of the Mothra larva cracks. This is because it is being damaged by Godzilla.
When Ichiro, Junko and Professor Miura first arrive on Infant Island, they were wearing yellow suits. After they were captured by the natives and brought to their chief, they were no longer wearing the yellow suits. The were wearing their civilian clothes underneath their yellow suits.
In the days before digital compositing, it was common practice to print matted elements with a degree of transparency, to minimize matte lines. It was an additional step of complexity to the process, which was deliberate and not an error. Many movies are notable for partially transparent mattes, including Robinson Crusoe on Mars (1964) and Star Wars: Episode V - The Empire Strikes Back (1980).
On Infant Island, there is an animal skeleton in the background. The head bobs back and forth, revealing that the body is a matte painting and the head is a prop hanging from the ceiling.
When the Mothra Larva first arrive on the island to fight Godzilla, the wheels underneath them to propelled them forward are briefly visible.
When Godzilla is moving the Mothra Larva up and down as it bites onto its tail, the skin of the suit actor, Haruo Nakajima, can be briefly seen through the pores in the neck. These are the pores in which a suit actor sees out of.
The first time Mothra flies away, the protagonists grab trees to stay upright. The large tree that two of the characters are holding moves and shakes unnaturally, revealing it is only a lightweight prop.
When you hear the warning bell sounding, it is the bell sounds like it's being hit continuously, but when it shows the person in the tower, you see him hitting the bell in single strokes, not continuously as is heard.
After the battle between Mothra imago (adult) form and Godzilla, the twin fairies start singing their song, with the main cast watching by. Then we cut to a military battle against Godzilla which takes place at night. Afterwards, we cut back to the fairies, who are still singing, and the main characters, who are wearing the same clothes as the day before. This seems to suggest that the fairies have been singing throughout the night and into the day, and that the human cast have not moved at all through out the night, as they are in the same position as before the military battle.