Geneva (I think?) was asking about an inexpensive setup...
this one? which I DO NOT OWN... looks reasonable - you get 2 mics with a USB mixer to plug in to either Windows or Mac for $30, $20 if you buy a paid membership on their sponsoring site.
http://www.thekaraokechannel.com/online#tag=18yvsz0iimg&page=membership_kit

HOWEVER there may be "just as cheap" options out there... this is just one.
IF you own RockBand for Wii etc or similar, that USB mic that comes with it WILL plug into your computer as well!

Lots of folks on the karaoke sites using a headset like the one you describe though!
OK good luck!!