I have to say that I share your feelings about e-mail, but what are the alternatives?
I suppose there are a few situations where emails are sent that you actually care about:
Someone else performs some action that you want to know about, so you can respond - posts an answer, turn, whatever.
You have forgotten your password and need a new one.
And actually, that’s about it. #2 can be circumvented by using a system like OpenID - essentially “make it someone else’s problem”.
#1 is nebulous and difficult - you could argue that RSS feeds are a solution, but to take a system like Stack Overflow as an example, I don’t want to have to subscribe to a feed for every question I ask/answer to get updates for it. Maybe you could have a ‘personal’ feed that updates only things relevant for me. Actually, I quite like that as a solution, not something I’ve seen (noticed?) implemented. Not quite so useful in situations where you might use a service once or twice though.
My Inbox is almost exclusively used as a notification area - things happen on one of the various websites I use, my iPhone beeps to tell me I have an email. I respond when I have a moment - all the while being reminded by my unread count that I have some action to take at some point. Often though, an email arrives that is just some boring marketing thing or a spam that got through or a chain letter from my mum, resulting in a false-positive beep from my iPhone.
RSS doesn’t quite cut this mustard - although some of that may be down to the way most RSS clients are implemented. It also doesn’t help for the idiots and old people who only have e-mail, but maybe we should start treating them like IE 6 users - just pretend they don’t exist and hope they go away?