But what is minor? Even if it was a misdemeanor and occurred many years ago, a drunk-driving conviction is not a minor traffic offense, for instance. A background check will definitely uncover it. If you didn’t fess up to it on the application, you'll look like you're were trying to hide it. And there's a good chance that could cost you the job opportunity.
Database searches of criminal records can be notoriously inaccurate. Arrests don't always lead to convictions. Charges get reduced.
One of the many unfortunate side effects of identity theft is that it can turn up wrong and damaging information during background checks. Good screening firms will always double check Internet search material with actual court records.
Unless you authorize an employer to search them, your education, military service and medical records are confidential. But these days, the authorization you sign may indeed include searches of those records.