You file as single, or head of household if you qualify.
Because you lived together more than half the year, if there are children under age 19 (or children who are under age 24 while being full time students), then either parent can claim the children as dependents, no special permission or release form is needed for this year. If you can't agree on who claims the children, the IRS will award the tiebreaker to the parent with whom the children spent the greater number of nights. (You may need to get a calendar and count the days. The IRS does not base their ruling on what a state court custody order says, but on where the children actually physically spent the night.)
One or both parents might be able to file as head of household if, considering the entire year, that person paid more than half the costs of keeping their home, and they claim a qualifying dependent. For example, suppose you each paid 50% of the household costs until October, then you each paid 100% of the costs of your separate household. In that case, each of you paid more than half their overall housing costs for the year. However, if one spouse paid 25% of the costs from January to October, and 100% of the costs of their separate household after that, it only works out to them paying about 43% of their overall costs for the year.
