Which one's correct? I've seen both claiming to be correct...
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'Faire' would is an old-fashioned spelling and would be somewhat pretentious nowadays. You also sometimes see 'fayre' in the context of food, normally something like 'We serve traditional home-cooked fayre'.
Thus one goes to is going to a modern event, it might be the state
Fair is the standard word in modern English, and is usually what’s appropriate.
Faire and fayre are older spellings, used in specific names but not widely used as generic terms today. So a fair might call itself The Cottesloe Village Faire to evoke historical associations; but except when referring to it by name, it would still usually be called a fair. (Just as one would write “Jayne’s Ye Olde Gifte Shoppe is a very nice shop.”)
There are a few exceptions: faire and fayre get used as generic terms within some historical re-enactment subcultures—most notably, for Renaissance faires. If someone writes “I’m going to a faire next weekend”, I would assume they mean something like that.