As a serious radio listener (60 stations from all over the world on my scan list; typically listen to at least half of them every day), my standards are high. I've been using InternetRadioBox on my iPod for three months, and am completely satisfied with it.
Internet Radio Box does what radio fans want it to do: plays livestream radio, allows you to customise station names and order in your playlist, and -- very important -- lets you enter your own raw stream URLs; you're not handcuffed to a preset list of available stations. It handles most stream platforms, including Windows-exclusive ones that even iTunes can't resolve. Yet the interface is easy and intuitive, quick to learn. Nor does it annoy the heck out of you with advertising or social media come-ons.
InternetRadioBox itself is very stable. It will crash every once in a long while, usually after a period of intense station-surfing. Just reboot and pick up where you left off. Again: this is rare.
If you take radio seriously, this'll set you up.