I always leave them on. I only turn them off if they annoy me enough that I feel the need to go into settings and disable them. At which point I would rather delete the app in most cases if they are spam (only did this to one game). Generally I will leave sound and alerts on, and turn badges off because I don't like the badges on my folders. But I always give the app a chance at push notifications.
Growl for iPhone. (Growl is a notifications app for macs and PCs) Looks like this: When I was jailbroken, I loved this thing. One of the main reasons I want to jailbreak. It does all notifications, SMS, push notifications etc... and stickies them, you can see a preview of the message, and tapping it will open the specific app for the notification like mail or any app. It's incredibly useful for gaming, as it will pop up but not pause the game you are in or interfere with the gameplay. unlike the apple popups that stop everything.
