James;

I actually used a black bed-sheet taped to the wall for this shot. I also had a bit of light spillage on the background and some of the detail was visible, so I simply burned these areas in post to ensure the background was fully black. If you are buying a background, just make sure it is a matte-type finish if possible. It will help keep it looking fully black if any errant light does hit it.

I can't see any reason why a constant light source wouldn't work in a softbox, but keep in mind that it would have to be quite bright or else you would have to eliminate all other ambient sources of light. Typically flashes/strobes are much brighter (although shorter duration) so that you can overpower the ambient if required and adjust your appeture/shutter/iso to ensure the only light which is captured in your photo is from the flash.

Stephen