They vary a lot by state. In California they're pretty weak. In NH, you can't have any perforation in your bodywork bigger than a quarter (about 2cm diameter) or it won't pass inspection. In New York you don't have to have an engine (only brakes, lights, solid suspension and frame as well as no cracked glass).

On a national level there are no laws about car upkeep, it's all done at the state level.

-Zeke