Do you know where it would show up on battle.net?

It will appear as one of your games there. It will be the Cataclysm box art with a small grey strip saying Beta under it.

There is no key that you'll get in your mail or anything. If you won the contests or got it through opt-ins, it will automatically appear as a game license on your account with all the other games you have access to.

Under "My Games" apparently.

You'll know you've been invited into the beta by two things: First, logging into your Battlenet account, under games management, you'll see the Cataclysm Beta, among your other games.
2nd, an email from blizz saying you're invited. Be aware that it won't be a plain looking email (I imagine anyways since I was in the SC2 Beta and it wasn't plain text).

But yeah my first bet would be to check your battlenet first.

As the original comment in this tree started... For most of the SC2 Beta entrants, we got our e-mail quite a while after the game was available on our account. I didn't get an e-mail for two weeks after I had been playing in the Beta. I actually got the "invite a friend to Beta!" e-mail before the one telling me I was accepted.

Send me your mail and pass I'll check it for you :3