Push apps that I use and recommend. Links all go to the iTunes store and I get absolutely nothing for recommending these apps.
Social Networking
NOTE: The free apps (Yahoo messenger, AIM, Nimbuzz) are not very good when you want reliable notification of your messages. You'll find that is true of all push apps - you get what you pay for. Push requires a substantial cost on the development side for reliable usage by thousands of users.
WhatsApp is a nice messaging solution for users who want to keep in touch with all their friends that have iPhones.
Facebook doesn't really need an explanation.
Beejive IM is perhaps my favorite IM client and it can keep you connected to your messaging service for up to 7 days!
IM+ is just as good as Beejive. It is more of a user preference on which one to get.
Twitter, eMail, RSS, Growl
Boxcar provides extremely timely push alerts, whether it's for email, Twitter, Facebook, or rss.
Prowl takes all your Mac Growl messages and displays them on your phone. For example, you can be notified when torrents complete with Transmission.
GMAIL
I recommend GmailPush, or you can pay another $0.99 for email notification from Boxcar. (I think it's $0.99 anyway). You would need to setup a forwarder in Google to forward your email to a special boxcar address if I remember correctly.
GmailPush works with your Google login/password and has excellent push from what I have heard.
It's up to you on which method you think is more secure. Either forwarding all your emails or giving out your gmail password to a third party.
Other apps
Notifications
The app has horrible ratings because the developer changed from a paid app to a free app with in-app purchase of notifications and every original customer dislikes him for it. The app itself is solid and has excellent response time.
It is reasonably priced and provides a REST api where you can actually send yourself pushes for anything, say if you wanted to add push notifications to a shell script on a linux box.
pushme.to is nice. Anyone can go to a website and send you a push (if you give them the URL).
Craigspro+ is a nice app if you are into Craigslist. You can set up searches and it will notify you when something new gets posted.
Sportacular is great if you are into sports.
Please share with others all the apps you recommend.
