Regardless of which parent is claiming which kid(s), as dependent(s), in any given year, the taxpayer must have a "qualifying person" living with him more than half the year (183+ nights) to claim Head of Household (HoH). So, it is possible for both parents to claim HoH, by scheduling where the children stay during the year.
For tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, for each child, is that the child live with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent.
The custodial parent has first priority on claiming the children on her taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree.
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household (HoH) filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the exemption to him.
So, it's good idea for parents to coordinate, so they both don't try to claim the same tax benefits, which will cause the IRS to send out letters.
