What if I am the non-custodial parent and I tried to exercise my visitation rights and the mother denies them.
My children are 12 and 9, I went to her place of residence and she denied my visitations. I called precint in her area and a Sheriff was called out. Even having my court papers he said he could not force the children to go. What am I to do now if no one makes her abide by the court order.
It is not the job of the constable or the Sheriff to enforce the visitation orders. Your answer lies in getting an attorney and filing an enforcement action against her in the court which entered the visitation orders. You may get "make-up" time, your attorneys' fees and the satisfaction of watching her get fined, possibly jailed, possibly placed on probation conditioned on her future compliance with visitation orders, possibly required to post a cash bond to insure her future compliance.
A warning to you--enforcement matters are complex and technical because they are "quasi-criminal". That means you need an experienced attorney to do this for you. Do not try to do it alone. You won't be successful. I'm not trying to alarm you, but I've defended contempt/enforcement matters where the attorney trying to enforce the order didn't know how to get it done.