I have just noticed a bug with the inbox improvements
I couldn't see this previously reported on this post
When you click on your inbox in a community you aren't a part of you get the old functionality inbox.
I noticed this from clicking a Hot Network Question which took me to a post on Home Improvement SE (not a member of this) and having a notification that prompted me to click my inbox. I thought maybe that the inbox improvements were still being rolled out in phases so I went to a more popular site Academia SE (again not a member of) and the old functionality persisted. The new style inbox works on every community I am a member of.
Here is a screenshot from my view from Academia (not member):
And my view Stack Overflow (member):
I also just tested this by going to Web Applications SE (not a member) - old functionality, joined the community and then the inbox changes to the new functionality.