In the US, businesses can legally reject employment based on an arrest record. In France it's illegal to post arrest records where there's no conviction; probably wanted posters are an exception though.
Well, in the US as a whole, businesses can generally reject employment for any reason, no matter how petty. There's a small, finite list of reasons that are illegal -- race, sex, and the like -- but these have to be specifically enshrined in the law.
Local governments can add their own laws on top of this, of course. Discrimination against sexual orientation has become illegal in many places within the US.
The letter of the law is one thing - the recommendations of HR lawyers to employers wanting to stay out of trouble are far more conservative. The advice I've heard is basically "don't do anything to create or discriminate against a 'protected class'". So even if it's not strictly against the law to decline to hire or promote people who wear blue shoes on Tuesdays, it's probably a bad idea to make such a policy without a strong, well-documented, objective basis for it.
I don't know if that's universally true in the US. Most labor laws such as these are set on a state (and sometimes local) level, so it's kind of a hodge-podge. For example, in Louisiana it's still legal to discriminate on the basis of sexuality, but not age, race, gender, or disability. But then in Orleans Parish (parishes are counties in Louisiana) it's not legal to discriminate on the basis of sexuality.
I'm not a lawyer though, and I have no idea what the federal law says regarding arrest vs conviction records. I do know that I've filled out numerous employment applications in many states that required me to disclose convictions of misdemeanor or worse crimes, but never any that have asked about an arrest record.
