Some YouTubers are claiming that falling star fragments only happen during a  full moon. This is false. This video has better information about finding those elusive shooting stars.

.https://www.youtube.com/watch?v=oeAtgxLKYf0

The game has an anti-farming system which seems to work on an eight-day cycle but I have found that each region has falling stars on different nights. Each tower region will get falling stars and is not dependent on the eight-day cycle of other regions. 

What ways have you found to get those shooting stars?

never was able to get one

I have about three. I'm afraid to use them. No idea what they are for.

There is one star fragment I got from a chest in a shrine. I can't remember what shrine that was, though. 

Upgrades for standard gear and amiibo gear. Also, you can use them in recipes.

The moment when I discovered you could actually catch the shooting stars was awesome.

I had 3 then I sold 2 because they gave me a lot of money for them. I keep one to make a recipe but I'm afraid of wasting it on something horrible.

I'm doing everything by myself. No guides, no videos, nothing. Currently at Rito Village to my second divine beast (and over 80hrs), I saw some shooting stars, but I was too lazy to go catch

You need them to upgrade the advanced gear to the final level. 

Then I think I'll keep them from now on

Good for you. I have the guide book for BOTW but I am staying out of it unless I get really stuck on something. I think I just glanced at the book twice so far. I have to be at ropes end before I use the book.