What is the ruling of Islamic law regarding a wife's refusal to allow her husband's mother to live with them, and her threat of divorce, and what is the most appropriate action in this situation, taking into consideration filial piety towards the mother and the stability of the family?

May Allah reward you for your eagerness to honor and care for your mother. We ask Allah Almighty to grant you her contentment, so that you may attain the contentment of Allah, the Exalted. If you can persuade your wife to agree to your mother living with you in the house, that is a good thing, and your wife ought to help you in honoring your mother. However, if she insists on refusing, then she has the right to a separate dwelling. If your house has sufficient space such that you can allocate an independent section with its own facilities for your mother, then there is no harm in that. Otherwise, seek a way to rent a house for her and provide a servant to help her and ease her loneliness, and make sure to visit her regularly and check on her condition. We also advise you to strive diligently to reconcile between your mother and your wife.