TuneIn Radio Pro. It's earned a coveted spot on my home page for plenty of good reasons.
First of all, there's the breadth. The app claims it offers access to more than 40,000 stations around the globe. I don't know the exact number but I can tell you I've never searched for something specific and came up empty. All the stations I care about are here...at least so far.
Next, and this is by far my favorite feature... I can record stations and listen to them later. Yes, it's like a DVR for radio. I never thought I'd care for something like this but I'm discovering regular radio programs that are on at inconvenient times, so I just queue them up via TuneIn's recording feature and listen later.
Be forewarned that there are some glitches with TuneIn's recording capability though. Every so often I'll set the app to record a station and it doesn't. I haven't been able to narrow down the culprit but it seems to be tied to whether TuneIn was the last app running before the recording is to start.
TuneIn Radio Pro is a universal app, so the 99 cents you spend on the iPad version also gets you the iPhone app. The iPhone app is even more quirky when it comes to recordings. You have to make sure your phone is plugged in or it won't record; that's not a problem with the iPad app as I've made plenty of recordings without plugging it in.
This one's a steal at 99 cents. Even if you don't think you listen to much radio today you'll discover a whole bunch of interesting stuff you've been missing out on...and you'll be able to record and listen when you have time. Highly recommended!