We don't use WebDAV much anymore (it isn't in Exchange as from 2010), but sometimes we still get cases on it. While there are many tools that can do detailed analysis and troubleshooting, I have a script that will test whether WebDAV is working as expected. I had to update this recently due to changes in IE10, and so I thought I would share it (before WebDAV disappears completely!).
The script is hosted in a standard HTML page, though as it is VBScript it requires Internet Explorer to work. Simply load the page in IE, allow the script (you will most likely get a question asking whether to enable active content or not - you need to answer yes), and then fill in the blanks (server name and mailbox name). When you click Go, the script attempts to read the top ten items of the given folder (by default this is the Inbox). You may be prompted for a username and password when you click Go, in which case enter appropriate details for the mailbox you are accessing.
That's it! A simple script to confirm WebDAV works.