That mostly depends on the definition of "big thing".
My experience with the WS stack and SOAP and all the acronym soup is that it takes an awful lot of workforce to deploy it. The status of the frameworks is complex, and definitely not something a hobbyist can put to work in a couple of afternoons. We have seen how many things on the net became the next big thing just because they were easy. Easy to understand, easy to interact with, easy in technology. Wikipedia, twitter, digg, youtube are internet big things, and they are, from the interaction point of view, light years away from SOAP/WS based interaction. They are KISS: simple and stupid. A whole horizontal market was opened just because of their simplicity. Even multiprocessing platforms like BOINC don't use anything near the WS stack, but they are the core of many high-throughput efforts.
Now, if you have to deal with complex multi-host transactions, authentication, credential delegation, caching... WS is there. It's the target that makes the need: banks, flight reservation, stuff like this. but they won't impact the common programmer. They require too much energy and too many different competences at once to become something usable for a horizontal market of developers.
Also, I am a REST person. I never advocated SOAP with much emphasis, but there was nothing else and it was a better evolution over XMLRPC (which, if you have to perform dumb RPC, IMHO it's still a good choice). Now I changed my mind. You mostly have resources on the web, and you interact with them with HTTP methods. SOAP is nothing but RPC on hypersteroids. No, REST is not the solution that replaces WS. At all. it's simply easier to use and to debug, albeit more difficult to design (you have to think in terms of resources instead of method calls). It's KISS. That's why it has more chances for success on the horizontal market.