It depends. If you are using TurboTax CD/Download, go to Forms mode, scroll to Form 740-NP and check the Military Spouse box.
Use the My Info section of your return to capture the military details. If you use the same home of record as the military spouse, then do not indicate that you lived in another state.
TurboTax: Military Spouses and State Taxes is a great resource to explain the Military Spouse Residency Relief Act (MSRRA).
During the state interview, if you had nonmilitary state income, then you may need to add Kentucky as a nonresident state. Only the Kentucky Form 740-NP for nonresident or part-year residents has a military spouse checkbox.
See 2019 Kentucky Income Tax Return Nonresident or Part-Year Resident Form 740-N
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