Most internet radio stations use progressive HTTP streaming (SHOUTcast/Icecast style) which can be played back within an <audio> element or Flash. This works well but can be hard to get right, especially if you use SHOUTcast servers as they are not quite 100% compatible with HTTP, hurting browser support in some versions of Firefox and a lot of mobile browsers. I ended up writing my own server called AudioPump Server to get better browser and mobile browser support with HTTP progressive.
Depending on your Flash code and ActionScript version available, you might also have to deal with memory leaks in creative ways, since by default Flash will keep all of your stream data in memory indefinitely as it was never built to stream over HTTP. Many use RTMP with Flash (with Wowza or similar on the server), which Flash was built to stream with to get around this problem.
iOS supports HLS which is basically a collection of static files served by an HTTP server. The encoder writes a chunk of the stream to each file as the encoding occurs, and the client just downloads them and plays them back seamlessly. The benefit here is that the client can choose a bitrate to stream and, raising quality up and down as network conditions change. This also means that you can completely switch networks (say from WiFi to 3G) and still maintain the stream since chunks are downloaded independently and statelessly. Android "supports" HLS, but it is buggy. Safari is the only browser currently supporting HLS.
Compatibility detection is not something you need to solve yourself. There are many players, such as jPlayer and JW Player which wrangle HTML5 audio support detection, codec support detection, and provide a common API between playback for HTML5 audio and Flash. They also provide an optional UI if you want to get up and running quickly.
Finally, most stations do offer a link to allow you to play the stream in your own media player. This is done by linking to a playlist file (usually M3U or PLS) which is downloaded and often immediately opened (as configured by the user and their browser). The player software loads this playlist and then connects directly to the streaming server to begin playback. On Android, you simply link to the stream URL. It will detect the Content-Type response header, disconnect, and open its configured media player for playback. These days you have to hunt to find these direct links, but they are out there.
If you ever want to know what a station is using without digging around in their compiled and minified source code, simply use a tool like Fiddler or Wireshark and watch the traffic. You will find that it is very straightforward under the hood.