I mainly use WinAmp on my Windows machine because it allows multiple instances of the program to run, allowing me to listen to more than one feed.
As Josh stated, it doesn't require much space on the screen, so even 6 instances of the program running can be tucked away in a corner. When the instances are reduced to their smallest size, the essential information about the stream is still visible.
I've also created a shortcut icon for every stream I frequently visit. I modified each shortcut to include the location of each stream. A single mouse click opens WinAmp and connects directly to the stream.
This eliminates the web browser entirely.