We've been seeing these a lot, and have determined that many of them come from Microsoft Office products. In particular, Microsoft Office.
See "How documents are opened from a Web site in Office 2003" for somewhat of an explanation.
I have been able to receive some brief relief by adding a mapping for the DefaultHttpHandler in web.config for those two verbs:
<add verb="*" path="*.mvc" validate="false" type="System.Web.Mvc.MvcHttpHandler, System.Web.Mvc, Version=184.108.40.206, Culture=neutral, PublicKeyToken=31bf3856ad364e35, processorArchitecture=MSIL" />
<add path="*" verb="OPTIONS, PROPFIND" type="System.Web.DefaultHttpHandler" />
This causes the "OPTIONS" request to succeed, and causes a "501 Not Implemented" status to be returned for "PROPFIND".
After 19 failed MS Word 2007 finally decides that it can use a "GET" request to retrieve the file, and that works (the file was acutally being served legitimately).
A little research shows that StaticFileHandler works even better for this. It returns 200 OK for both OPTIONS and PROPFIND verbs, along with what appears to be valid data, as long as the request is targeting an actual resource. When Word probes the folder itself, this returns a 404 Not Found.