When a relationship ends, the housing decision often depends on several things happening at the same time. The property may need to be valued, equity may need to be divided, debts may change, support payments may need to be established, and each person may need to qualify for housing on their own.
That is why it helps to review the mortgage options before assuming the home has to be sold or that a traditional refinance is the only way one person can keep it.
A useful place to start
Decide what each person hopes to do with their housing first. Then the mortgage, equity, and qualification can be reviewed around that plan.