Here's Bram in 2004 arguing that live streaming is a niche market: http://bramcohen.livejournal.com/4294.html
In the comments, "streamerp2p" is discussed, which I used as the basis for a proposal for live radio station streams along with Speex, back ~2005, when bandwidth was expensive. It's still around, apparently: http://www.streamerp2p.com/
Bittorrent itself can do streaming of static content: just request the blocks in order. It can also do streaming of live content if you chunk the live content and have an out-of-band way to update the torrent you're looking for, but that's not great.
In my proposal I mention some other systems as well:
Peercast: http://web.archive.org/web/20060207013338/http://www.peercas... (defunct, Internet Archive link)
P2P-Radio: http://p2p-radio.sourceforge.net/ (defunct)
Allcast: http://web.archive.org/web/20060805040005/http://allcast.com... (defunct, Internet Archive link)
Chaincast (dead and blocked from IA)
Abacast: http://web.archive.org/web/20061017041905/http://www.abacast... (IA link, they're still around but now cloud-based)
Xiph's IceShare: http://wiki.xiph.org/IceShare (defunct)
Robert Haarman's StreamDist prototype: http://inglorion.net.nyud.net:8090/software/#streamdist
Andrew Brampton's research and prototype: http://web.archive.org/web/20100325135652/http://www.lancs.a... (IA link)
Onion Networks' Swarmstreaming (defunct and blocked)
The proposal isn't worth posting any more; all the assumptions and business cases are obsoleted.
Today, rather than use a client at all, I'd probably push for WebRTC P2P support, funding patches in all the browsers and their mobile versions, incentivize people to upgrade their browsers, host with cheap bandwidth for everyone else, and just hold on until enough of the market has upgraded.