You can't file a joint return unless you are legally married. You would file separate returns. Most of the time, the person who pays more than half the family living expenses should file as head of household and claim the child as a dependent, and the other person should file single with no dependents. That usually results in the lower overall taxes. There are different combinations that are possible, if you wanted to know exactly what was best you would have to try them all on different test tax returns.
Just remember that only one of you can claim head of household, because it is impossible for both of you to pay "more than half" the household expenses. (If it is exactly half, no one can file as HOH, but the person who pays more than half only has to pay $1 more than half, if it is fairly close.)
And, only one parent can claim the child as a dependent.
you can't file a joint return if you are not married.
at best one of you can claim head of household by including the child as a dependent on your return. The other files as single,
who should claim the child? you need to do a series of returns one where he claims the dependent and head of household. you file as single.
1) if both returns show refunds or balance due add them if one shows a balance due and the other a refund net the two
do the same where you claim the child and head of house and he files as single.
repeat 1 bove
the filing option that produces the smallest balance due or the largest refund should be best.