You can find out if a car has a rebuilt title by ordering a vehicle history report from a data company like AutoCheck or CARFAX. Before ordering, you’ll need the vehicle identification number (VIN) in order to process the report. The Department of Motor Vehicles can also inform you whether a car has a rebuilt title. However, this information is only available to the owner of the vehicle. Furthermore, the DMV can only inform you of the brand on the title (salvaged, rebuilt/restored), not the reasons why it was branded so. A vehicle history report provides ownership, location of registration, accident and repair history.
For more info, see A Rebuilt Car Title: Pros vs Cons