We were able to track down the issue and resolve it on several sites.
The problem appears to have been with the template framework that we use on all sites.
There was an update that was automatically applied overnight.
This update appears to have had some conflict with some other extension which caused the main page of the site to fail.
We've not been able to identify the exact extension conflict (it only happened on a few sites), but have tested all sites and the latest patch of the template framework resolved the issue and has been applied to all sites we manage.