Is this covered by Apple's warranty?

I've had my iPhone 5s for approximately 4.5 months. About two months ago, I notice a hairline scratch on the screen. I didn't really notice it that much, so I just forgot about it. Recently, though, the scratch seems to have gotten worse. I now notice it more often, especially when looking at something light in color (mainly white). Will Apple give me a replacement for free? I read that hairline cracks are covered under Apple's 1-year limited warranty. I have this phone for the next to years and I just don't think I can live with the scratch on the screen. I have not once dropped my phone or put the screen near something sharp.

A scratch is no a crack. And scratches are not just caused by "sharp" objects, but they are caused by things harder than the glass that gets scratched. Also you noticed this scratch 2 months ago, but not 3.5 months before that. So if I were the Apple employee, I would offer repairing the screen, vs replacing the phone. Hopefully they will make a one time free screen replacement, and you telling the truth and being polite will go a long way into getting it repaid. Jim is correct here, so just let them know the facts and do not add your emotions or how you think it got scratched. The folks at the Apple stores have pretty much heard it all, so ratchet down the emotion/drama, and just ask for your options.

Just so you know, scratched are not covered under the normal Apple Care warranty, but would be covered under Apple Care+ warranty.

There's no way to know unless you take it to an Apple store and have them check it out. Apple leaves it up to their employees to make the judgment calls so be polite and ask kindly, they are more likely to cover it if they don't feel you are attacking them.

If it's physical damage. No. Unless you have insurance on your phone then you're probably outta luck. If it's a surface scratch, you can probably get a filler to make it disappear. You could probably buff it out too if it really annoys you. Apple will give you much more information than I, so it's worth a call!

Applecare only covers defects, not damage caused by you. A scratch is not a crack is damage and not a defect. Best chance is make the Genius Bar appointment at the store, explain the issue and beg. Just maybe they will replace the screen for a few bucks. But thinking you will get a new handset is out of the question. Best luck.