If you're female, you may already know that clear nail polish brushed onto a ripped pantyhose can stop the nylons from ripping further. If you don't already know that, then you're welcome. (But let's all agree to never wear pantyhose and make that professionally acceptable.)
Here are some other uses of clear nail polish that could be useful in an emergency situation:
Bug bites: Clear nail polish might stop a bug bite from itching. According to CSI (yes, the TV show), it works for chigger bites. If you don't know what a chigger is, just be thankful and store this away in your long-term memory in case you ever need it. According to Lifehacker, other people in the interwebz say it can stop itching in any bug bites. If I didn't have any hydrocortisone cream nearby, I'd definitely try this. HT: Lifehacker and Bella Sugar.
Liquid Bandage: Whether you call it a liquid bandaid or new skin, clear nail polish accomplishes the same things for a fraction of the cost. HT: Instructables.
Secure Your Glasses: Have a loose screw? In your glasses, not your head! I assume you have a screw loose upstairs. But maybe you don't wear glasses. However, maybe you wear sunglasses? If the screw tends to be loose (or you're just proactive), dab some clear nail polish on top of the screw to help seal it in. You could use crazy glue, but the risk is too great that it'll end up where it shouldn't, and then you'll have glasses that can't move. HT: Real Simple.
Temporarily Repair Broken Glasses: If you somehow crack your lenses (I've never seen anyone do this in more than 20 years in the visually-challenged community), you can secure the pieces together temporarily with clear nail polish. This should be enough to get you home to your back-up pair (you have one, right? Try Zenni Optical.) or to the optometrist for a new pair. HT: Lifehackery.
Secure Your Buttons: Have a button that's starting to get loose? Get proactive and put some clear nail polish on it! Or you could rip it off and re-sew it. ...Or more likely, you can wait until it actually falls off and creates a regrettable clothing malfunction. Even if you would re-sew it, perhaps you won't have time for a few days. Clear nail polish buys you time. HT: Hackaday and Bella Sugar.
Secure Other Stuff: Like the glasses trick above, you can use nail polish to secure any screw just a little bit more. Paint polish on the screw and screw it in before the polish dries. HT: Instructables. You can also use it to secure things that aren't life-or-death such as model figures and costume jewelry stones. HT: Lifehackery.
Protect Your Labels Make labels smudge-proof by adding a layer of clear nail polish. HT: Instructables and Bella Sugar. This is an especially good idea for medication labels. HT: TipHero.
Thread a needle: Dip the end of your thread into clear nail polish to make it sturdier, and thus, easier to thread through the eye of a needle. HT: Instructables.
Color Code: Have identical things? Distinguish them with a bit of nail polish (you'll want to use a colored nail polish). While commonly used for keys and kosher kitchen utensils, this can work on everything from golf balls to flashlights to clear toiletry bags. HT: Instructables and Lifehackery.
Mark Levels: Same as color coding, but for a different purpose. Mark a bucket or spray bottle's "fill level" with colored nail polish. With different colors, you can mark different levels, such as one level for cleaner and a second level for the water to add. HT: Instructables. You can also mark your preferred levels on the shower, thermostat, or radio. HT: Homesessive.
Waterproof matches: Brush some clear nail polish on the head of the match to protect it from moisture. HT: Bella Sugar and Lifehackery. Learn how to do this correctly at Outdoor Life or WikiHow. You can watch a comparison of normal matches, waterproof matches, and nail polished matches at YouTube. (Watching the entire video may make you want to claw your eyes out. This should have been done in 3 minutes tops, not 10 minutes.)
Mark Household Poisons: Who needs a jolly roger when you can use colored nail polish to draw a big, fat X on your household poisons? HT: Homesessive.
Prevent Splinters and Snags: Coat splintered wood to keep it from snagging you or your clothes. HT: Bella Sugar and Michelle Phan. Especially useful on pretty wooden hangers. HT: Homesessive.
Smelling Salts: If you run out of smelling salts for your damsels in distress, you can use nail polish as an alternative. Apparently ammonia (and thus, cat urine) is also a good choice. HT: Lifehackery.
Other useful ideas:
Prevent costume jewelry from turning you green (aka, tarnishing). HT and HT. Can similarly make your metal belt buckle shine. HT.
Prevent further fraying of a window screen. HT.
Fill scratches in a wood floor. HT.
Smooth a broken mirror edge. HT.
Shut holes on your salt shaker if it pours out too much salt. HT.
Seal scuffed shoes. HT.
Stop a windshield crack from spreading. HT. Remember that a cracked windshield is probably ticket-worthy in your state, so get that repaired. It's also easier and cheaper to fix a small crack than waiting for it to get large enough to be a nuisance.
Wart remover. HT. Guess that's why wart remover always smells like nail polish...
Want to get really creative? Imagine the things you could do with glow-in-the-dark nail polish! HT: Lifehackery.