Older kids, as @empress-eremmy mentioned do seem to get some say legally, but unfortunately younger children not so much. Here in the U.S., the mother seems to get the children by default, and almost has to be found unfit by the courts for this not to be the case. There are numerous cases where it is clear to anyone with eyes the mother is unfit and she still gets the children.

There needs to be more punishment for people who level false allegations (and not just regarding family court). Years ago, a friend of my ex wife had begun leveling allegations against her husband who had left her that he was sexually molesting their 3 year old daughter. I had never thought it true. He was gone almost a month before he went back and no further mention by her of such horrible crimes were said again. So either she made it all up out of spite, and he came back to avoid never seeing his daughter and to avoid possibly going to prison for her lies. Or he was a monster and she was willing to sacrifice her child to be with him. Neither scenario gave me much hope for the little girl who was a pawn either way. Either way, she was a monster.