My son's father has not been in regular contact for over 3 years. Although he is named on the birth certificate, what rights does he have?

Sounds like you haven't gone to court for support, custody and visitations. What rights do you want him to have? What rights does he want? Does he pay you anything for the child?

The same rights as any legal father. Lack of contact doesn't change that.

If you have not went to court and got a custody order, he is a equal parent with you at this point as a parent listed on the birth certificate

I don't want him to have any rights as my son is settled and happy. His father is unreliable, he has been in and out of prison, has a known violent temper, and has threatened to kidnap him on more than one occasion. He has not paid anything towards him since he was born.

What you want and what the law allows are two separate things. To terminate his rights you have to go to court and courts are very reluctant to grant a TPR. Generally they will only do so to clear the way for an adoption or if the parent represents a danger to the child. You might, therefore, get a TPR on the latter grounds, but you have to prove he's a danger.

But it sounds to me like he's not exercising his rights. So why go to the expense of trying to get a TPR? If there is no court ordered visitation, then you can refuse to let him have contact. That would force him to take you to court where you can present your case against him.