There is a really cool new Azure App Service Web App tool called appLens that is explained in more detail here. Open it by selecting the Settings –> AppLens from the App Service you are interested in analyzing, similar to that shown in Figure 1.
Looking at the output of the report I see that there was some some down time on 3-JUN-2016 at around 00:30. I then decided to take a look at the Event Viewer logs accessible via the KUDU console which I discuss here and here. And what I found is shown in Figure 2.
And the Warning event 2299 logged just after the one shown in Figure 2 further proves the root cause. The detail for event 229 is: Worker Process requested recycle due to ‘Status Code’ limit.
I also took a look into my IIS logs and see the 500s recorded, then after the restart all is back up and running. I might actually fix this one day…