A married taxpayer must always file as married. Either married filing separately, or married filing jointly. Sometimes head of household is allowed, but only if the spouses have lived separately, and the taxpayer cares for a qualifying dependent. (A spouse is never a qualifying dependent for HOH, and HOH is never allowed if the two spouses live together at any time in the last 6 months of the tax year.) You may want to review the dependent and filing status rules in publication 501.
For 2021 and 2022, your client's options were to file as married filing separately (the wife would not file unless she had income, in which case she would also file as MFS and obtain an ITIN), or your client could have filed as married filing jointly by applying for an ITIN for his wife. Single was never an option. Note that when filing MFJ, your client would include all his spouse's income, even if it was earned overseas and subject to tax overseas. Also note that if your client lives in a community property state, half his income should normally be assigned to his spouse when filing MFS.
It sounds like your client needs to file amended returns for 2021 and 2022 to change from single to MFJ or MFS. Yes, your client can amend to MFJ using the new SSN. Because the SSN was not issued prior to the filing deadline (April 18, 2022 or April 18, 2023, I believe), the wife may be ineligible for certain tax benefits, such as EIC, even if they would otherwise qualify.
In fact, going back before 2021, you don't say when the client married, but as you must know, a US citizen is required to file and pay income tax on all their world-wide income, no matter where they live. If the client married before 2021, they would also have been required to file as married for those years (2020, 2019, etc. however far back the marriage goes). Again, either MFS, or MFJ with an ITIN. I can't evaluate the consequences of failing to do this without a lot more information.
