You federal tax refund is never taxable income.
A state tax refund, it may or may not be taxable income on your federal return.
If you took the state and local tax deduction last year and itemized, the state refund will be taxable this year if the deduction benefitted (reduced) last year's tax bill. This is true even if you opted to apply the refund to this year's state or local return.
But if you took the sales tax deduction or the standard deduction last year, your state refund won't be taxable on this year's return. It also won't be taxable if last year's return was subject to the AMT (Alternative Minimum Tax) and the refund amount was less than the amount disallowed under AMT.
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