Nonresident Spouse

You cannot claim Head of Household if you do not have a qualified person as a dependent.  A spouse can never be claimed as a dependent.

See this TurboTax support FAQ for Head of Household when married - https://ttlc.intuit.com/turbotax-support/en-us/help-article/taxation/married-person-claim-head-household-filing-status/L078u1RSW_US_en_US

If you are a US citizen or US resident and your spouse does not have a Social Security number or an ITIN and you are not applying for an ITIN with the tax return then you can only file your tax return as Married Filing Separately.  Where asked to enter the spouse's Social Security number enter 999-88-9999.  You can only print and mail your tax return, it cannot be e-filed.  When you print the tax return erase the Social Security number for your spouse and manually enter NRA for non-resident alien.

See this TurboTax support FAQ for the procedure to print and mail a tax return using the online editions - https://ttlc.intuit.com/questions/1944348-how-do-i-print-and-mail-my-return-in-turbotax-online

If you decide to file your tax return as Married Filing Jointly you must apply for an ITIN with the tax return and you would need to be providing a statement with your tax return that you want your Nonresident Alien Spouse Treated as a Resident.  See IRS Publication 54 Tax Guide for U.S. Citizens and Resident Aliens Abroad pages 7 and 8 - https://www.irs.gov/pub/irs-pdf/p54.pdf

Go to this IRS website for ITIN information - https://www.irs.gov/individuals/international-taxpayers/general-itin-information

You cannot file as Head of Household because you do not have a qualifying dependent.  Your spouse can NEVER be a dependent, and you say you do not have children.  Sorry---paying to support your spouse who lives abroad does not make you "HOH" for tax purposes.

Am I Head of Household?

https://ttlc.intuit.com/questions/1894553-do-i-qualify-for-head-of-household

https://ttlc.intuit.com/questions/2900097-what-is-a-qualifying-person-for-head-of-household

If you qualify as Head of Household, when you enter your marital status (single or married filing separately) into MyInfo, and then enter your qualifying dependent, TurboTax will offer HOH as your filing status.

If you are married, your choices are to file married filing separately or married filing jointly.

See this for further information:

https://ttlc.intuit.com/community/taxes/discussion/married-joint-with-foreign-spouse-no-ssn/00/2857757

How did you file in the past?  You may need to file amended returns for those prior years as well.

As said, you can't be head of household if you are married and living with your spouse.  HOH is a special tax status for single parents, it does not mean "head of the family".  You can file married filing separately or married filing jointly, using the instructions as described. 

@w1vet , assuming that you are a US person ( in this case a citizen/ GreenCard ), a GreenCard application/sponsorship ( I-130 ) takes a significant time -- INS quotes approx. 37 months.   You could visit the nearest  INS office and get an idea of issuing period -- and if they are waiting for some thing from the country where your  spouse resides.   I am just wondering why you did not ask for a K-visa  for the spouse -- but that is in the past.

I will not comment on your filing status options as my colleagues @Opus 17 , @xmasbaby0  and @DoninGA   have already covered the subject admirably.