Question says it all: there must be one that stands a head-and-shoulders above the rest...which is it? I've got a few bits of software that would like to store syncable data on a WebDav server...
Apache's mod_dav has worked well for me.
I would suggest taking a look at: http://www.ajaxfilebrowser.com/. No registration is needed it will automatically create a folder for you when you access it first time.
You can give http://dropdav.com/ a try. It enables to access a dropbox account with WebDAV.
you can also try 4shared.com http: //webdav.4shared.com:80/wa you can get 15GB for free But the webdav works only for WinXP or Linux Link to get it working http://www.4shared.com/faq.jsp#q36
Update: box.net does not offer free WebDAV anymore, sorry. You should try other providers. I use http://www.cloudsafe.com - they offer 2GB of free online storage and free WebDAV access can be easily enabled for every safe.