That would depend on whether the spouse was an NRA at any point during the tax year.  If so, your client, for the purpose of HoH filing status, is considered unmarried and may file as HoH so long as the home which he/she maintained was the home of the new born at least for half the time since birth.

Otherwise, the default filing status is MFS.  Given the spouse was a resident alien at the end of the year and had no income, it would be more beneficial to elect joint filing unless there are other concerns such as joint and several liability.

ITIN or SSN will be required if your client is to file MFJ.  For other statuses, it's not required but the return will need to be filed on paper.