Spent the morning debugging a Sitecore CMS 6.2.0 (rev. 091012) install on my local machine. Most of the site worked fine, but things started to fall apart under the admin screens.
The Content Editor UI would come up, but the Presentation > Preview resulted in a 404.0 exception where it was trying to resolve the Request URL using the StaticFile handler to files that didn't exist.
The admin Desktop interface was similarly out of order. In this case bringing up any of the UR elements would result in a 404 using the StaticFile handler again.
The resolution I found was to change the Application Pool from Classic Managed pipeline mode to integrated. The exact cause still eludes me but I suspect there may have been a custom URL rewriter involved that wasn't intercepting requests with the classic mode.