What is my 2018 Tax Filing Status? I'm a permanent resident, married, and live in the U.S. My spouse chose to live separately from me and has never been in the U.S.

If you are a US citizen or permanent 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