Visible crew/equipment: In the scene when Bedford hits the ceiling in the Cavorite-covered chair, the thick metal cable attaching the chair to the ceiling is visible to the right of Bedford. It is NOT, as I initially suspected, part of a lamp or some other bit of scenery - at one point you can actually see it move as it pulls Bedford up and down.
Other mistake: The sphere operates through the opening and closing of Cavorite blinds to negate the force of gravity. It has no engines at all. Why then, when it lifts off from the moon, do you see a blast of what appears to be rocket exhaust?
Continuity mistake: Watch carefully as Bedford falls through the crystal prism into the underground city. On the surface he has his helmet on but in the very next shot, as he plummets down, he has no helmet. He explains that it must have 'fallen off' but you do not ever see it doing so. As the helmet would have to be twisted to be released, and no such action took place, it cannot have come loose.
Other mistake: Because there is no air on the moon, Cavor advises Bedford to touch helmets when they speak, otherwise they will not be able to hear each other. Yet there are several times where they seem able to communicate without doing so.