I'm continually amazed at how the premier Web properties are willing to share what they are doing. We get to peek behind the curtain routinely. Google and Yahoo both have good lab pages, but there's some seriously experimental stuff on the Mozilla labs page. Here's what they're up to.
Snowl
The latest lab project, Scowl intends to make Web apps look more like their desktop equivalents, and is focused on messaging. It's more than just an Outlook replacement, though, and includes RSS feeds, Web forums, social networking feeds, and other content pulled into the API. An alternate view on all of your messages and feeds looks more like a newspaper or something like NetVibes with only text.
Weave
Getting this whole "sync" idea right has proved a major challenge. You can use Google Bookmarks, or skip that and just feed your links into Del.ici.ous. It's still a pain. Mozilla would like to solve the problem with Weave, which syncs things like your bookmarks, tabs, and even Internet history. You can download Weave and test it, but be ready for crashes and bugs galore.
Prism
Web apps live (obviously) on the Web, but they don't really have to. Prism is another experimental endeavor where Mozilla has made it possible to launch an app by itself with just the app wrapping. Let's say you use a Web 2.0 site for your accounting; you can launch just the site as an app that uses its own memory space. If Firefox crashes, the Prism app would continue unabated.
The Coop
This visual link sharing system is a bit like Twitter in that you can see what friends are doing on the Web, with pictures of them to help you track their activities. You can click the links they are visiting. You can also share videos, photos, blog posts -- any kind of rich content you come across.
Personas
Xbox 360 gamers will immediately recognize the concept behind Personas -- it is a "lightweight" customization add-on that lets you give more personality to the browser, akin to the Xbox's custom faceplates. I think they could take this much farther and allow users to really customize the entire experience, not just the header portion of the browser.