This includes not only a visual inspection of all safety-relevant parts of a car but also a check of the emission values.
How is this handled in the US?
Source: I have an appointment for car inspection this morning.
Edit: certain counties
There are exceptions with older (vintage) cars, and relatively new cars.
As with most things, these laws vary from state-to-state, and even county-to-county within said states.
Kentucky had a limited emissions inspection regime from 1999-2005, when it was killed by the Republican governor. The only current inspection requirement I know of is if you purchase an out-of-state used vehicle. We have plenty of ancient cars on the road that have never been inspected.
Sometimes police officers will pull you over if you have obviously-broken headlights or such, and tell you to fix it, but not often.
en.wikipedia.org/wiki/Vehicle_inspection_in_the_United_States