Nail polish spills on your fabric and blankets can be incredibly frustrating, but they aren't actually all that difficult to get out. Nail polish remover is the best place to start, but you have to be careful that the remover doesn't stain the blanket as much as the nail polish did, so don't use colored nail polish removers.
Spot test your nail polish remover in an inconspicuous corner of your blanket to make sure that the remover itself won't stain or make the blanket's colors run. Do this by gently rubbing a clean rag dipped in acetone remover over the surface of the blanket. You should be fine if the color doesn't bleed onto the rag or change the shade of the blanket.
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Stretch the blanket taught across your lap or over your sink, as this will make it easier to apply nail polish remover without spilling it all over everything. Gently wipe the stain with a nail polish remover-soaked rag, being careful to use a clean spot on the rag once it has absorbed the nail polish color. If the blanket has absorbed a lot of the nail polish, you may have to turn it over and repeat the process on the other side.
Once most of the nail polish has been removed, prepare a bath with a mixture of half hydrogen peroxide and half water. Rub some gentle liquid detergent directly into the stain and then rinse it out in the bath. The hydrogen peroxide will lift out the rest of the stain that the detergent has broken up.