I have a 6 yr old daughter.Paternity hasn't been established.Does the father have any rights? He did sign the birth certificate when she was born.

This can be a very specific state issue,

In some states even if DNA latter proves he is not the bio father, if his name has been on the birth certificate for so many years, he is the legal father and can not get out of his legal obligations even though he is not really the father.

He can or if there is doubt, you should get a DNA test to find out who the father really is, And he as the father can do that anytime he wants, if he does not have custody, he can get a court order for it.

But the father has any right to custody, visitation and the such.