Here's a sobering fact: I can't think of a surface I haven't spilled nail polish on, multiple times. Past victims include clothes, furniture, rugs, and a tortoise. Being a human lacquer hazard, though, I've picked up a bunch of ways to vanish vernis, even when the situation seems dire. And if you allow me to share my hard-won knowledge, I will feel much better about my lack of nail-eye coordination. Will you help me, help you? Please? OK then, I'll quit babbling and make with the solutions. Here's how you get Chinchilly and Zulu off the following items:
First rule of Clean Club: Never, ever, ever, ever use nail-polish remover on finished wood. It will literally take off the finish and leave you with discoloration much worse than a little spilled polish. No, what you want for this job are mineral spirits. Pour them on a rag and then wipe them over the polish until it's all gone—very simple and effective. Just wash the area thoroughly afterwards so there aren't spirits sitting there, soaking into your floors or chair.
Use acetone! Clear acetone, specifically, since a lot of polish removers are dyed. The most important thing with carpet is not pushing the polish in deeper. So if it's still wet, gently place something absorbent on top—paper towel is great—to soak up as much as you can. Then it's time to dab. Don't pour acetone directly on the carpet; put it on a rag, then gently dab it onto the stain, over and over. This will feel like it takes a million, bajillion years. Maybe you should hire someone off Craigslist to do it. Or, consider it an exercise in mindfulness and appreciating boredom—so zen! Just keep dabbing 'til it's all gone.
Clothes and Upholstery
If the polish is still wet:
Use a paper towel to absorb as much as you can (just lay it lightly on top). If the damaged item is a blouse, make sure none of the polish seeps through to the other side of your shirt by layering paper towel under the stain. Then, use polish remover the same way you would on carpet. If the polish smear is small, try using Q-Tips instead of a rag.
If the polish is dry:
As a first step, try holding an ice pack on the patch of polish. The cold should make the lacquer harden to a point of brittleness and contract a little so it's easier to break up and scrape off. You can also try picking the bits off with a pair of tweezers or brushing them gently off with an old toothbrush (if it's in tiny flecks). After you've gotten off as much as you think is possible, use the acetone Q-Tip or rag-dabbing technique up top.
Note: Is your item made of acetate or another lab-created fabric? DO NOT USE POLISH REMOVER; IT COULD DISSOLVE YOUR CLOTHES. Take them to a dry cleaner instead—they have access to a bunch of solvents we lesser mortals can only buy in the sketchiest corners of Amazon.
Acetone can melt plastic, so if you can't chip the polish off by hand (don't gouge at the surface of your phone with scissors or something, ok?), there's this stuff called Goo Gone that really works. It's composed primarily of toluene, acetone, and methanol, though (all nasty stuff), so use it extremely sparingly, no huffing it, and don't get any on yourself or anyone else.
Ugh, I'm sorry—there's no great solution here. If anyone in the comments has a miracle fix, please do tell. Otherwise, I just try to minimize by dabbing non-acetone remover on the stain and trying as hard as I can not to spread it or get the remover on unaffected pieces of the item. If it's dry already, try the cold-pack trick you use on fabric and see if it'll shatter off...?