I know, I know! Pick me!
In all seriousness, despite the party line generally being something like "If they're unwed, Mom is the only person with any rights to the child until the court says otherwise even when paternity is established", this doesn't hold true in most states. One of the "M" states - I think Missouri - actually does make it clear that in an unwed situation even if Dad has signed the AOP he has no right to keep the child until and unless there's a court order and I vaguely recall a Legal Aid site explaining what the single parent - yeah, usually Mom - can do, and it can turn into a criminal offense very very quickly.
But that's not the norm, despite the party line.