Is should've a word? It gets underlined in red in Google Chrome and Word.
Yes Should've is a word indeed. but its the short form of Should Have.
I often use this word in MS Word but it never gets red underline.
It does get underlined in Google Chrome, but one of the suggestions to correct it is would've, so it's just an oversight on their part. They also underline Google, so their spellcheck does need some work.