I bought a case immediately after buying my phone, but screen protectors were $40 at Target, so I just kept the plastic sticker on mine.
Unfortunately, it recently came off when I put my phone in my pocket.
The other day my phone slid out of my pocket while I was laying under my shed and the screen had scratches.
How far did it fall? Maybe an inch or two?
I started researching and everything is conflicting:
Mashable: Toothpaste, ultra-fine sandpaper, magic erasers, baking soda, and other methods work. I have used most of those for polishing headlights, except instead of removing whatever may be left of your UV coating you are removing the oleophobic coating.
I would be much more concerned with thinning your phone screen than your headlight.
JerryRigEverything shows an expensive machine that removes a thin layer of the screen along with minor scratches and then he replaces the oleophobic coating.
Cnet had a similar list to Mashable, except it sounds like she actually tested it. Allegedly a pencil eraser also works and she tried coconut oil, petroleum jelly, a banana peel, and automotive scratch remover. Those didn't work.
WikiHow shows using toothpaste\baking soda on phones with plastic screens and glass polish on glass screens.
Popular Mechanics allegedly tested some of these, but didn't share pictures or a video. He taped off the vulnerable areas of an iPhone 3g and, after trying other methods, used a buffer, and overheated the screen.
Lifewire mentioned other methods, but suggested something that I had wondered: Super Glue. You work it into the scratch and use acetone to remove the excess once it dries.
I don't know where I got the idea. I keep visualizing someone using super glue to fill a crack in glass and I people saying to use it to fill chips in windshields, but I visualized using a new razor to scrape off the excess.
Techwalla mentions other methods, but the new one was liquid bandage.
I also wondered about windshield repair kits. I found several videos, including by JerryRigEverything, but people pointed out they didn't do anything to remove air trapped in the resin.
Can cellphones survive vacuum?
The Action Lab put an iPhone in a vacuum chamber and it still worked.
About the fifth video I found was this guy who used a generic version of a kit that ChrisFix showed where you use a syringe to pull air out of the chip.
He didn't like it and said that his video with super glue worked out better, but he put a bad spiderweb in his screen: