It depends on whether or not you are legally separated. If you are legally separated, then each of you file as single, or, if one qualifies, as Head of Household. However, if you are not legally separated (or divorced on December 31, 2016), then your only filing options are Married Filing Joint and Married Filing Separate. Neither can file as Head of Household because this requires you to have been living apart for at least the last six months of the year, and you separated in July. Here is an FAQ that provides more information on this: https://ttlc.intuit.com/replies/4206137
If you file Married Filing Separately, you both may lose good credits for your return, which includes the Child Care Credit. I personally have worked with separating/divorcing couples in this situation who decide to do one last return filing joint to maximize their return, and then work on an agreement as to how to divide it up. If you and your soon-to-be ex can do this as well, you will probably get the best tax result for your situation, but you are not obligated to do so.
If, however, you are legally separated and can thus file as either single or Head of Household (now the last 6 months rule does not matter), you can claim the credit if you are the custodial parent, the one with whom your child lived the longest. Since with your situation you lived together longer than 183 days, you each could have a claim, but the tiebreaker goes to where the child was the longest, even if by one day. However, if it is an absolute tie (possible with the 366 day year), then the parent with the highest AGI gets the claim and all of the benefits if they choose.
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