I thought Wave sounded amazing when it was first announced. When I finally got an invite to it, I have to admit, I was deeply underwhelmed with the user-facing implementation. It's probably the slowest, least responsive, web app I've ever used. It also felt way over engineered, with far too many unrefined ideas crammed into it.
That being said, I ended up being a big fan of it, hoping Google would eventually get around to fixing some of the obvious issues (they mostly didn't, though it did get slightly more responsive before it died).
Why was I fan? I didn't think of Wave as a thing to do, like Twitter (you tweet), or Facebook (you friend and post), or a blog (you write a journal)...there never was a way to "Wave", just like there isn't one way to "e-mail". It was just a new set of communication tools that you could assemble however you wanted.
It's no different than using e-mail to plan a party in one case or send photos to your mom in another case or applying for a job in another. E-mail is just a tool, the use-cases fall out of how you use that tool.
For some reason Wave seemed to look like a solution in search of a problem, but I ended up finding several existing things that were better solved in Wave than with the existing workflows. The most notable was organizing a group around collaboratively writing a large document. Structured correctly and the work just kind of "fell out".
The mistake with trying to use Wave was always in trying to use all of the features in Wave.
I really miss it and am very glad to see bits and pieces of the idea slowly percolating into other Google products (Drive for example).