While I would admit that there is little to attract me to the idea, I already know of one internet based application that friends of mine are finding very useful.
My best man at my weeding is a licensed plumber. He uses Intuit's Quickbooks to help keep his accounts straight. Since he has an accountant, he uses the internet version of Quickbooks and no longer has to transfer files to his accountant. They both can log into a central site and the latest information is available to either of them.
They've been using this setup for about two year now, and it seems to have worked well. I can see where, if there was a need to share information with multiple people or computers/devices, an internet based application might be well suited.
Frankly, I still expect the bulk of my own computing to be based on my own hardware, but when I think of how much time I spend on the internet I have to ask, "aren't we already sort of doing this anyway?".