A little bit of history of the web makes me think of a clear path of evolution that I see this way:
1. The Static Web
This was the first expression of the web and it was doing all it was created to do. The web was a way of hyperlinking documents and that's what we had originally. It already had a great potential, the concept of a growing shared network was powerful in itself. The light protocol was appealing. The glorious simple tagged language spreaded rapidly. Page indexes were the search tool of this era.
2. The Dynamic Web
3. The Social Web (known as Web 2.0)
This is the era of user as a content provider. Here were born most of the current collaboration tools: wikis, blobs, social nets, resulting in collaborative, powerful communities.
Sometime during this web, the monolithic web era finished and the web started to have little interactions in what was eventually named as ajax. This flexibility was a turning point, this was a great moment in the establishment of the web as a platform, as it was acquiring one of the main characteristics it's competitor had to their advantage (windows desktop environment). At this point the concept of thin client is without a doubt the way to go.
As far as search the tools (almost tempted to use this word in singular) of this era are search pages with intricately secret formulas and interesting revenue models exploiting the inefficient search in a disorganized information world.
4. The Relational Web
This is the web services web, rest web, the mesh ups web, pipes web. This are applications talking to each other and collaborating to make information more integrated and more meaningful to the end user. There's also a turning point in the device expansion at this point where the concept of computer is more irrelevant and replaced by the net concept. Any device that gets me into the net is basically taking the place of what was the computer before. Here is where we're at right now.
Not sure about search evolving much in this era ... maybe as pipes are built we see more vertical searches in different tools for different domains.
5. The Semantic Web
This web is the web of meaning. There will be some type of common understanding of some type of relevant meaning outside of languages and countries specifics that will make the web an infinite source of knowledge. We'll go from having information to having "Thoughts". It really will work increasingly similar to querying our own brain, except for one little detail: interfaces. The search will change dramatically as we can search by meaning in a web that is meaning enabled. I doubt the current search revenue model will hold.
I'm amazed to see that the browser as a tool almost had no evolution along the first three eras, it might be limitations in the protocol itself that dis-allow the browser to evolve ... but I doubt it, I believe it's more circumstances that caused this.
Another thing to notice is that web evolution is accelerated (it'd be nice to quantify how accelerated it is, some more date facts should be added ...).
Obviously, we have to wonder ... What's next? New interfaces that enable us to "think" in the net: our "extended collective brain".
We should expect most probably a change in technologies. We all love the web, but as bad as it sounds, we have to think of its death some day. It already went far from reaching its own limits. Plus, the media soon will have changed too much, the speeds will have changed, 3D worlds, audio and voice will have evolved so much that we'll need a new protocol that supports that better and was created with this world in mind. Also to be considered are all the advances in medicine and in human-machine integration that is making us increasingly cyborgs.
So, I'm thinking mostly about user interface changes, our need to communicate with "the computer", which at this point is losing so much relevance yielding to became more "the net", is going to be totally different and will need native protocols that support it. So then users can extend them to promptly (why can't you say fastly, if you have prompt and promptly? some day we'll have a consistent logic universal language too ;) reach its own limits again and again ...