People have been arguing about this FOREVER and I thought I'd clear it up:
Yeah--yeah is just slang for yes and doesn't sound anything like "yea" or "yay". The h at the end of the word gives it the short a sound.
Yea--yea is the formal way of saying yes. Like in the Bible and The Book of Mormon it will say yea for yes and nay for no.
Yay--yay is a joyful exclamation. Yes, I think it should be a real word, even though a lot of people don't think so. (No it's not in the dictionary, so I guess it's not an official word. Work with me.)Anyway, yay is what you say when you're excited or happy. Like, "Yay, we're getting cookies!" You wouldn't say, "Yeah, we're getting cookies" or "Yea, we're getting cookies" because those are answers to someone's question. Yea is too formal for me, and I think yay just looks cuter anyway.
Hope this helps!
P.S. For those wondering, "ya" is text talk for "yeah", not a real word. :)