If court finds extraordinary circumstances, then the court looks at what is in the best interests of the child. A petition for visitation is filed in Family Court, usually in the county where the child lives. All parents have a right to visitation with children unless the court determines that visitation is harmful to the child. Family Court is supposed to grant liberal visitation to parents. Grandparents and siblings also have a right to petition for visitation of a child, even after the parents' rights have been terminated, and may get visitation if the court determines it is in the child's best interests.
Visitation orders can range from unsupervised weekend visits to supervised day visits.
The fact that the father's name is on the birth certificate does not establish paternity. If a father wants custody, visitation or to ensure rights once the mother dies or for any other reason, he must marry the mother, acknowledge paternity, or file a paternity proceeding in Family Court. The parent must go to family court and fill out a petition. Once the petition has been filed, the parent will get a court date to come back. The court clerk can explain how to do this.