FStream 4+

Works fine in my iPhone 3GS. Been using this app for two years. I've added many streaming URLs into my list and I'd like to be able to sort into folders but at least I can reorder them. I've used the record feature once only but never had an issue like another user reported. Would be five star review if I could put different streams into folders. If you're having trouble getting it to work, make sure the URL will work elsewhere (try safari or another app) before blaming this app. I've had the odd trouble but it's because the URL was wrong.

I'm running this on my iPhone 4 with IOS 4.2.1. This app is awesome! Exactly what I was looking for!

Pros:
1) Understands pretty much all formats of streams used on Internet Radio
2) Allows you to decide whether to allow or to not allow playback over cellular network. Very useful as my wife sometimes forgets to shut the radio off before taking the phone from the dock and putting it in her pocket. So I just changed "More->Cellular Network = OFF" and that's it - the radio shuts off automatically when she is no longer within the reach of our home WiFi. At the same time, we can still play it even over cellular network if we need to (say, the kids are getting fussy and want to listed to a story, etc) - in that case we make a conscious step to enable cellular playback and keep track of time to avoid AT&T surcharges (we are on 2GB/month plan).
3) Playback continue in the background! I can turn my favorite Internet Radio on and then go on with my life of reading and replying to emails, viewing presentations, etc etc.

Cons:
1) No way to customize the home screen - can't change the background, skin, or anytthing like that.

Highly recommended!

2020 EDIT: Classic and still works great as ever. Bravo, FStream team!

I've been using this app for over a year - originally on my 2G iPod Touch and now on an iPhone 3G. This is the best audio streaming application available on the iPhone or iPod touch, AND it's open-source! It comes with a great pre-loaded list of Internet radio stations that you can add to your favorites list. My only complaint is that by default, it doesn't switch smoothly from a 3G data connection to Wifi, or vice versa. However, with a little fiddling of the settings (switch buffer drop to 20%), it will work just fine! Now, I can walk around my college campus, seamlessly switching between Wifi and 3G while continuously listening to Internet radio. Thanks, FStream!