I have sort of the opposite issue of most other people. My failing verbs are GET and POST. PUT and DELETE are allowed for some reason. Here's my related web.config.
<system.webServer>
<handlers>
<add name="ExtensionlessUrlHandler-Integrated-4.0" path="*." verb="*" type="System.Web.Handlers.TransferRequestHandler" preCondition="integratedMode,runtimeVersionv4.0" />
</handlers>
<rewrite>
<rules>
<rule name="static dist files" stopProcessing="true">
<match url="^(.+)$" />
<conditions>
<add input="C:\Source\PRISM\DEV\PRISM\Service\PRISM.Service\dist\{R:1}" matchType="IsFile" />
</conditions>
<action type="Rewrite" url="/dist/{R:1}" />
</rule>
<rule name="index.html as document root" stopProcessing="true">
<match url="^$" />
<action type="Rewrite" url="/dist/index.html" />
</rule>
</rules>
</rewrite>