You can file as head of household if you meet all of the following requirements.
1. You are unmarried or considered unmarried on the last day of the year.
2. You paid more than half the cost of keeping up a home for the year.
3. A qualifying person lived with you in the home for more than half the year. A child is a qualifying person, even if the person claiming head of household can not take the child as a dependent.
4. One child can only qualify one person as head of household, so you can not both file that way.
If the two of you provided all the support for your son, then the one who provided more than half the support qualifies for taking the exemption.
Therefore, if one of you pays for more than half of the cost of keeping up the home, that one can claim the head of household status. If that person also pays more than half the support, that person can also take the exemption; however, if the other person pays more than half the support, that person can take the exemption.