If you are a U.S. citizen married to a nonresident alien, your default filing status is "Married Filing Separately" unless you qualify for a different filing status. You may qualify for the "Head of Household" filing status if you pay more than half the cost of keeping up a home for yourself and a qualifying child or other relative.
Regarding claiming an exemption for your foreign spouse, for tax years beginning after December 31, 2017, and before January 1, 2026, personal exemptions have been repealed, so you cannot claim an exemption for your spouse.
However, you have the option to elect to treat your nonresident alien spouse as a resident for tax purposes under IRC § 6013(g). If you make this election, you can file a joint return and your spouse will be treated as a resident for the entire tax year, which allows you to use the "Married Filing Jointly" status. This election is made by attaching a statement to your joint return, and it applies to all subsequent years until terminated.
If you do not make this election, you must file as "Married Filing Separately" or, if eligible, as "Head of Household."
Sources:
Publication 501 (2023)
Publication 519 (2023)
§ 6013. Joint returns of income tax by husband and wife
IRM Part 21. Customer Account Services. Chapter 8. International. Section 1. IMF International Adjustments
Publication 1380
IRM Part 3. Submission Processing. Chapter 38. International Tax Returns and Documents. Section 147. International Notices
