No one can really tell you the best way to file without examining your family financial information in detail. Here are some things you need to know.
1. You child can be the dependent of you, the other parent, or whichever grandparent the child lived with for more than half the days since she was born. (But, the grandparent can only claim the child as a dependent if both parents agree and if the grandparent has more taxable income than either parent.)
2. The child can only be the dependent of one person. Any taxpayer who is not claiming the child as a dependent should not even list the child on their tax return.
3. "Head of household" filing status only applies if the taxpayer claims a qualifying child dependent AND paid more than half the total cost of the home or homes where they lived and cared for the child. Based on your store, it sounds like you or your partner paid more than half the total cost of living in all the different homes you lived in, so neither of you can legally file as head of household, no matter who claims the child.
4. If the grandparent who could claim the child, only has income from social security, and doesn't also have income from a job or pension, they won't get any tax benefit from claiming the child as a dependent.
In the end, the strategy for the largest tax refund or least tax can only be determined by testing the different combinations. You and your partner both file as single, either with a dependent or without. Or, a grandparent claims the child, possibly allowing them to file as head of household. In the past, the optimal strategy was usually to have the person with the highest income claim the child, but that may no longer be the case. Anyone who does not have at least some income earned from working, is unlikely to get much tax benefit from claiming a dependent. But if you, the other parent, and the grandparent all have income from working, then it really can't be guessed without testing all the combinations. You might want to see an experienced tax professional.
