Visually indicate notification urgency level
I think it would be helpful to (perhaps subtly) indicate what level of urgency a notification has. I.e. make the user aware of the fact that app X believes that some notification is critical information (or less critical information). It would not only be helpful as a visual aid to people that get plenty of notifications and might be used to dismissing a lot of them quite fast, but it would also make it clear if/when an app is abusive and sending unnecessary "urgent" notifications that e.g. can't be blocked by the "Do Not Disturb" button. In those cases, users would be made aware and would, therefore, be able to voice their opinion to the application creator and so forth.
How would you like it to work
Perhaps an icon, a word, a color/border, or something like that would work well.