Lived in KY 01/21-05/21 collecting CO unemployment. Moved to NY 05/21 stopped collecting unemployment, have W2 for NY wages 05/21-12/21. Which state taxes do I file?

It depends.

You would file a part-year Kentucky resident return for the time you lived in Kentucky (1/21-5/21). All unemployment compensation earned as a Kentucky resident is subject to Kentucky income tax. Therefore, you would be taxed on your unemployment in Kentucky and any other income during your period of residency. 

You would file a part-year New York resident return for the time you lived in New York (5/21-12/21). You would be taxed on any unemployment received while living in NYS and your New York W-2 wages. 

Lastly, whether you have to file a Colorado return depends on where you physically earned the income for the Colorado employer. Per the State of Colorado, unemployment compensation is Colorado-sourced income if the employee was physically present in Colorado when performing the work. If that is the case, you would file a Colorado nonresident return for 2021. If the unemployment is Colorado sourced, then Colorado will also tax you on the unemployment. In this case, Kentucky (and New York if applicable) will give you a credit of taxes paid to another state. 

Please complete the Colorado state return first out of your three state returns so the calculations of the credit are accurate. Then, I recommend you complete the return for Kentucky, followed by New York. 

For more information, please see How do I file a part-year state resident return?