Can a single mother ask the child's father to sign over his parental rights?

Answer 1

Yes. You can ask for anything. Assuming the real question here is, can a single mother cause a "termination" of the father's parental rights, though, the answer is more likely, "no." In every jurisdiction I've ever heard of, this would require a court hearing and a finding that such a termination would be in the child's best interests.

The courts don't like to sever the relationships between parents and children because children need their parents, and parents have an obligation to provide for their children. Terminating the relationship means ending the obligation to support, too. Also parents have important constitutional rights to their relationships with their children. All in all, it takes something extreme and unusual for courts to terminate the relationship between parent and child.

In England parents do not really have rights in relation to their children but the law relating to children states that a child has a right to a relationship with both parents (the question of whether one parent is unsuitable is something to be thrashed out in court on an individual basis). However, if the child has not yet been born you can seriously restrict the father's ability to have any control over the child or contact with them by placing "Father Unknown" in the relevant box on the birth certificate - doing this even when you know who the father is is not illegal. This effectively means he has no say at all over what happens to the child. If you put his name on the certificate you are legally obliged to consult him over every non-trivial matter of your child's life (education, religion, medical treatment, moving house and so on).