To claim head of household you have to have a qualifying dependent and pay more than half of the household expenses where you live with that dependent. Only one of you can contribute "more than half" of the household expenses. If you have lived together with the child since it was born, then either the one who contributed "more than half" claims the child and HOH, and the other single with no dependents; or the one who contributes less than half takes the dependent but claims single, and the other parent also files single.
Note 1, it is not who earns more, but who pays more than half of your household expenses. If the person who earns more saves it all, or is paying off loans or other child support or tuition, then they might not actually be paying more than half of your current household expenses.
Note 2, sometimes neither one of you pays more than half, if you also receive outside support such as from family or the government (SNAP, section 8 housing, AFDC, etc.).
If it's a close call, you will actually have to add up your expenses and look at who pays for what. Household expenses include rent or mortgage, utilities, food, furniture, repairs, and similar expenses, but does not include medical or education expenses because they are specific to a specific person.
