Is 90's or '90s correct?

" '90s " or " 90s " , but never " 90's " , I would suggest. An apostrophe following a noun denotes ownership, or is a shortened form of "-is".

Numbers should be treated just like this.

e.g.

John's gloves are blue.

- apostrophe for John; he owns the gloves no apostrophe for "gloves"; there are two of them. The '90s are years in the decade ending -90, so it's a plural.

so for '90s,

I like the music of the '90s.

The 90s' music is the best.