There is no conclusive list at the moment, but generally speaking:
CARP Events (config sync errors, VIP state transitions, etc)
Filter reload errors
Gateway failures
Corrupt config.xml
Inability to upload a config with AutoConfigBackup
Basically, anything that makes a notice flash in the upper right, plus a few other things.
The notifications will be quite a bit more flexible in the future but probably not until after 2.2 ships.