This is an acronym that means "Go Screw Yourself". This is commonly used in text lingo.

when you're hungover as shit after a spending a night drunk as a skunk

Go Screw Yourself, made known by Avery (the singer).

When you try to type gay but it ends up as gsy, because you type fast, but carelessly, so that what you think you typed was 'gay' turned out to be 'gsy'

Being a nice dude who everyone genuinely likes.

GSYS stands for "Go Screw Youself",

you would say it when you feel angry about someone who is a douchebag, in general, for asses.

A terrible song by female singer "Avery", and it can mean "Go screw yourself" Probably most appropriate used on someone ignorant or annoying to get them to go away or leave you alone.