If a custodial parent doesn't have a stable place to live, what can happen?

You don't say which end of this you are on. If you are the custodial parent then perhaps you need to let the child stay with the stable parent for awhile while you get yourself stablized. If the other parent is less stable then you are then just do the best you can as fast as you can. For the child's sake......kids need a sense of security. Which means knowing where home is and that they know where they are going to be sleeping that night. Best wishes.