who keeps rental house?

if a married couple decide to separate and they are living in a rented house who gets to stay in the house?
no lease or agreement was signed. they just pd rent and moved in. sometimes she writes the check from the joint account - sometimes he does.
if asked, the owner would probably say that he considers the house rented to the husband as the husband spoke with him about the house and occ. does work on the side for him.
she wonders, if the occasion arises that she asks her husband to leave, will the police instead tell her that she has to leave, saying it is his house?the location is tennessee if that makes any difference.
thank you.