Right now in Utah it depends on the age of the car. Cars that are less than two years old do not require a safety or emissions inspection. After two years they are inspected every other year. After six or seven years they are required every year. 1967 and older do not require inspection.
Next year however, safety inspections will no longer be required but emissions will still be. I guess law makers figure that it is not okay to pollute the air, but it is okay to cause a wreck and possibly kill someone with a poorly maintained vehicle.
Some things checked are: mileage, proper window tint, cracked or broken windows, mirrors, ball joints, tie rod ends, shocks, exhaust, brakes, wipers, horn, lights, signals, tires etc.
Cost is $15-$20 for safety, emissions vary per location but average about $25-$35. It is state mandated but each county has different restrictions, ie rural counties usually do not require emissions.
Vehicles registered as antique do not require inspections.
One of these days I would like to take the Model A in and see how well they can perform an inspection on it. I'm also curious as to what the emissions results would be.