In Minnesota if a couple were never married but the father's signed the birth certificates can the mother legally move the children out of the state if she allows the father visitation?

Children should be able to see both of their parents when they want. If the parents are fighting that can be hard on a child. Unless a judge specifically orders that the father cannot see the kids, there shouldn't be a problem, no matter what state you live in. I hope you are not caught in a bad situation between your parents. In all US states the law presumes that the mother retains sole custody of children who are born outside of marriage unless a court rules otherwise. That being the case, the mother could move out of state with the children if there is not a shared custody, visitation, or other order from the court. The father can if he chooses contest the action and request the court to prevent the move. If the father has been active in the children's life and/or is paying support whether court ordered or voluntary and/ or if the couple resided together the majority of the children's lives, the father has solid grounds to stop the move and petition for custodial rights. If the father is in agreement it would be in the best interest of both parents to have the terms in writing along with other pertinent matters (visitation schedules, finances, etc.).