"Legally," they can get whatever is on the public record if they want to pay someone to do the digging.
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If they're looking for your driving record, then it's likely your job requires you to drive for the company frequently. If so, I wouldn't be surprised to see them pull a full history.
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It is really up to the employer, and how aggressive they want to be in "digging" up information on their applicants. Public record is just that, public record.
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