It depends. However, the fact that he does not have a social security number does not change your filing status if he was present in the U.S. for at least 6 months out of the year. You are considered married for tax purposes and should file either Married Filing Jointly or Married Filing Separately. Unless you lived separately for more than 6 months, you cannot file as Head of Household or single. He is not considered your dependent, but either a co-declarer or an independent declarer (if you file separate returns).
Having said that, there are some limitations when a spouse has an ITIN versus a social security number. The biggest of these is that a married couple cannot claim Earned Income Credit (EIC) when either spouse files with an ITIN. Likewise, EIC is also not available to anyone using the filing status of Married Filing Separately. For this reason, it is still typically more advantageous to file Married Filing Jointly. Please see this FAQ for more information: https://ttlc.intuit.com/replies/3288477
For more information on filing a return using an ITIN, please click on link for this IRS website: ITIN
What if a citizen and non documented person married in December? Would that count as having living apart for six months?