Although I rely heavily on email for my work, Twitter has definitely become a mainstay in my system tray. I use Twitter -- the service -- from a variety of locations and devices. From my desktop machines, I use TweetDeck. I like the way that the application places each of my feeds into columns.
When I'm on my iPhone, I simply use the Twitter app. I used to use TweetDeck on my iPhone but began to have stability issues with the app. So I just switched back to the native Twitter app, which I've found to be perfectly viable for my needs.