Snapchat is an app I found quite nefarious with push notifications. They have 'screen on' as the default notification mode for all their messages.
I currently use two phones with notifications turned off for Snapchat. However, the notification settings get reset if I use it on my other device. I don't use Snapchat altogether, but the other day, they sent me an SMS mentioning that I have a few unread messages. So scummy and so stupid!