Not while it's on a salvage title. It has to be repaired, inspected, and retitled as rebuilt first.
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In most states a vehicle on an active salvage title cannot be registered, plated, or driven on public roads. The brand exists precisely to stop a written-off car going back into traffic without anyone checking it first. What you can normally do is own it, store it, transport it on a trailer, and work on it. What you generally can't do is drive it away. The route back onto the road runs through repair, a state inspection, and a new title -- usually called a rebuilt, reconstructed, or restored title depending on the state.
The inspection is not a roadworthiness test in the MOT sense, and it isn't a quality review of the repair. Its main purpose is anti-theft: confirming the car is the car it claims to be and that the parts used to fix it weren't stolen. That's why the paperwork matters so much. Expect to produce receipts for major parts, the original salvage title, photographs of the damage before repair, and often photos taken during the work. Some states also check lights, brakes, glass, and airbags. What's checked varies, and the fee and waiting time vary with it.
A great deal. States differ on what triggers a salvage brand at all, who is allowed to buy an unrepaired salvage vehicle, what the inspection covers, what the retitled car is called, and whether the brand ever comes off. Some states restrict salvage purchases to licensed rebuilders or dealers. Some issue a rebuilt title that carries the history forever; a few use different language entirely. Because of that spread, treat anything you read online -- including this page -- as orientation rather than instruction. Your state DMV is the only authority on your car, and the rules do change.
The expensive mistake is buying a salvage car, repairing it, and only then finding out what your state required. Check the requirements before you buy: whether you're allowed to purchase an unrepaired salvage vehicle as a private individual, what documentation the inspection will demand, and whether the previous owner actually has the salvage title in hand. Then photograph everything as you go. The single most common reason a rebuilt inspection fails or stalls is missing paperwork -- not bad repair work -- and photographs and receipts you didn't take at the time can't be recreated afterwards.
Most states require insurance to register a car, and most insurers won't write road cover for a vehicle that isn't road-legal. That's a loop worth planning around rather than discovering. In practice the sequence usually runs: repair, pass the inspection, receive the rebuilt title, then arrange insurance, then register. If you're buying a car already on a rebuilt title, all of this has been done -- it can be insured and registered like any other used car, subject to your insurer's own view of rebuilt vehicles.
In most states it can't be registered or plated, and driving an unregistered car on public roads is an offence. You can generally own it, store it, work on it, and trailer it. Rules vary by state, so check your own DMV.