Twitter is here in the Windows Phone Store, but until this is fixed you might want to consider an alternative - Twabbit is a favourite (though stay clear of the tweet streaming option for battery reasons), or try Mehdoh.
Thanks to those writing in to report this, I'll update the story when the situation is resolved. I wonder how fast a fix can be rolled out and pushed through Microsoft Q&A? I guess we'll find out!
[Update] The Twitter client is now working again, though the underlying possibility of a total crash must be something the developers are looking into. Watch this space for an app update. Anyway, whatever was changed on the back-end has now been reverted or fixed, it seems. Panic over!]
Interestingly, you can't use the mobile Twitter site either, tapping on the mentions tab gives: