I found three that seem very interesting, but unfortunately only one works well on my computer.
EyeOS
eyeos is the Open Source Cloud Computing’s Web Desktop. However, eyeos can be useful to you in different ways depending on your needs.
EyeOS can run on your local computer, a local server or you can create an account and run it on their free server eyeos.info. It can therefore be used by schools or companies as an intranet or you can use it privately or with friends.
Out of the three webOS this one works best for me at the moment. Here you have everything you need to create your own office on the web, there is a pop3/imap email application, plenty of space for word documents on its servers, a feedreader, calender, spreadsheets, internal messaging, an ftp client etc.
It furthermore has a text editor and a good word processor based upon OpenOffice.org Writer (version 2.4 at the moment). A new word processor with extensive collaborating tools is on the way.
The new EyeOS 2.0 will be released on the 1st. of January 2010, many screenshots of the new applications are already available on the EyeOS blog.
Its Word Application looks promising, with an emphasis on collaborating. You can read all about it in this EyeOS Blog post and here is a screenshot:

Cloudo Beta
With Cloudo, every computer, in school, at work, at your friends’ or even in the library becomes your own, free of charge computer. And with Cloudo Mobile your online computer is accessible from your mobile phone as well.
Use Cloudo as your default computer, or use it side by side with your work or home computer.
When logging in there is a clear warning: ” Please Note - Cloudo is still far from being ready for prime time”. Users are invited to try out the beta version for a limited time
It already looks beautiful however, but it is indeed not very functional yet. There is an application called Word that probably one day will be the word processor, but at the moment it actually is a text editor. There is for example no option to change the colour of the selected text or to insert a picture.
There is a application called Textpad, which must be the text editor, as the default courier-font also seems to hint. Here I can change the colour of the text, but only of all text. No support for pictures either.
It does have a functioning email application with your own address username@cloudo.com and for example a feedreader, calender and contact manager.. Cloudo is an open platform made to make it possible to add or even develop your own applications.
And by the way… I think that clock is just perfect.
icloud

icloud doesn’t start in my Firefox browser but it is also clearly stated on the icloud homepage that Firefox is at the moment only supported in an early alpha state. So I have to be patient.
I opened it briefly on another computer in windows with Internet explorer (see screenshot), but I wait with a more thorough look at it until I can open it on mine. One of the benefits of a WebOS is that it can run on different Operation Systems, so I suppose they will soon update the Firefox handling.
It looks very promising though, see for example this page with its features explained.
There is an application called Write and I read that they provide an email account as well: username@icloud.com. It offers 3GB of webspace.
Related Post:
G.ho.st a Web Operating System with Zoho and Google docs integration

