When Battra first makes landfall, as the terrified Japanese pedestrians are running for their lives, take note of a woman to the right of the screen wearing a brown pant suit and carrying a black purse. She is still running in the next crowd scene on the sidewalk. The next time we see her, she is running up the stairs from the subway, having apparently taken the "A" train to escape the rampaging monster.
When the larva Mothra bites Godzilla's tail in the ocean, look closely at the base of the tail as it's tossing Mothra upwards. You can clearly see the tail is a separate prop on some kind of support that has distorted it's shape - widening it unnaturally.
When the squadron of jets attacks the larval Battra as it is heading for land several of the flares used to simulate fired missiles visibly bounce off the surface of the water.
When Godzilla is grappling with Mothra Larva in the sea, right before he throws it, a string is visible on Mothra.
Strings holding up Mothra and Battra are visible in some scenes. The bright studio lights make them easy to spot against the dark background.
After the earthquake is detected at the Mt. Fuji observatory The Cosmos begin singing to summon Mothra back to japan their mouths aren't moving when they recite the first "Mothura" of the song before pausing and moving to the nearby window to continue singing.