I have push notifications enabled, but think that the game does not tell give notifications for the most important events. Currently, the the game notifies you when: 1.) your world energy is full, 2.) a new titan spawns, 3.) titan loot is available, 4.) a building upgrade finishes, 5.) crafting finishes, 6.) your base has been raided.
I find the first four quite useful, am indifferent to the fifth, but think the sixth is useless. I honestly couldn’t care less when I’ve been raided. In addition, there are a number of other things I with the game would send me a notification for that it doesn’t. When: 1.) my alliance energy is full, 2.) my raid energy is full, 3.) a mystic vision is available, 4.) my free daily summon is available, 5.) a war or challenge event has begun.