There are soooooo many radio apps for iOS that it takes some effort to get my attention. But airadio pro has done that (yes, the app name is all lowercase). This free app is an advanced internet radio tuner that also records what you might want to save and lets you listen later or send them out via email as an attachment.
The app contains links to about 8,500 stations, and they are grouped by genre, or you can search by name. The app supports multi-tasking so you can grab the station you want and do other things while the music plays in the background. You can create a favorites list, and "time-shift" streams by pausing or rewinding.
The app supports MP3, AAC, AAC+v1 and HE AACv2 streams. If a call comes in, the stream is paused and will resume when you are done. You can even select your choice of VU meters, from traditional physical-looking ones, to newer LED bars.