Usually when a court gives one parent sole custody and the other either no visitation or restricted visitation or supervised visitation, it would be very detrimental to the children and the court with great reluctance will in fact grant one parent sole custody and the other supervised visitation or no visitation at all if there is substantial evidence that the children may be endangered or not taken care of by the non-custodial parent who may have certain impairments. The grounds that the court usually looks at, among others, is mental illness and substance abuse and neglect.
Sometimes it may be necessary to hire an expert to interview the children and to evaluate both parents for a determination of which parent should have custody. Usually this is accomplished by both parties agreeing to have an independent psychologist perform this evaluation and that usually saves a lot of time and cost if the parties cannot make a determination between themselves.