Fingernail polish stains are tough to remove and can only be removed with a chemical. Removing these stains takes some patience and skill. The best product to use is an acetone based fingernail polish remover. You should test the acetone on your carpet before you try this process. Good test spots are in a closet or somewhere that is not noticeable. When testing, place a small amount of acetone on your test area and let it sit for a few minutes. Then remove the acetone by blotting it out with a white towel. If there are no imperfections made from the acetone onto your test spot, you’ll now want to treat the stain directly.
You should use a dropper to apply the polish remover to the stain. Let the chemical soak, but not too long (most acetone based polish removers evaporate quickly). You should only let it sit long enough so that the polish starts to run. Then blot it with a white towel. The fingernail polish should lift off onto the towel and stain should be lightened. If the stain remains try the process again. Your last resort is to make a trip to your local store where they sell products that may work better. Be sure to follow the instructions carefully. With these types of stains I recommend contacting a professional carpet cleaning service.