The Disney vs Sony court case decision (aka The Betamax Case) ruled that it was indeed legal to copy broadcast television for one's own use and not for resale. Radio broadcasting, LPs, CDs without copy protection also were ruled to be legal to copy for one's own use. This is called the Fair Use provisions of copyright law. The Fair Use provisions do NOT apply to copy-protected content, such as DVDs or CDs with copy protection schemes, because what is illegal is circumventing the copy protection.
Streaming audio is considered similar to radio. If you have, say, Audacity running with the "Stereo Mix" or "Wave Out" input selected, and you're playing an audio stream, you'll be fine as long as you save it for your own use and do not sell it to anyone.
It is just like recording off the air on radio or TV.
Tom