I spend a great deal of time surfing SE sites from my phone during commutes and other down times. The global inbox is quite effective at reaching me even for chat pings. Quite frankly the mobile site is very well done!
However as a community moderator, there is a whole class of information not getting through. There are no notifications for moderator flags! I can switch to the full site and see these but that is inconvenient for browsing, so I only end up checking for flags periodically.
Ideally the moderator tools should all be available in mobile templates, but as a stand-by it's reasonable to switch to the full view to handle flags, but flags would get addressed in a much more timely manner if I knew about them. As it stands I often come and go from the site without even knowing there are outstanding flags.
A simple flag counter added to the header next to the notification counter and linked to the tools page (which automatically opens the full version anyway right now) would fix this.