MVC routing fails and IIS 7.5 returns 404 with StaticFile handler

I had a new build Windows 7 machine with IIS7.5 and encountered an old problem again.
An MVC Web API application I had installed was failing to run and IIS was returning 404 and was indicating that the StaticFile handler was being executed:

The handlers were configured correctly in the web.config:

Running aspnet_regiis -i against the correct .NET version didn't do anything.
People recommend that you don't add the <modules runAllManagedModulesForAllRequests="true"> element to the web.config, so I didn't. What was going wrong?

In the end I found my own blog post! The problem was that the ISAPI filters were disabled by default.

The problem was solved by turning them on.