First of all, the space shuttle probably breaks the sound barrier 10 seconds after leaving the lauch ramp (if not sooner). Secondly that thing you see surrounding the tip of the tanks and the nose of the shuttle, is just air being compressed at high speeds, even airplanes do that sometimes, specially with their wing tips.
And the last part, is just the extra tanks being jettisoned.
So this video is NOT about the space shuttle breaking the sound barrier.

50 seconds or so into Discovery's (STS-114) launch you will see a vapor or condensation cloud which has all the appearances of a Prandtl-Glauert condensation cloud -- a transonic cloud caused by the Prandtl-Glauert Singularity.

http://fluidmech.net/tutorials/sonic/prandtl-glauert-clouds.htm
http://www.unexplained-mysteries.com/gallerynew/displayimage.php?pos=-953

Sorry EMPIRE but during this video is when the sound barrier is broken. You don't hear it, and it has nothing to do with the end of the clip with the SRB jettisoning. As pho3n1x points out those are Prandtl-Glauert type clouds which only occur from about .95 to 1.05 the speed of sound. The vapor trails you are speaking of are created by low pressure areas above wings or on trailing edges.

http://www.stemnet.nf.ca/CITE/sts_ascent.htm

"By 45 seconds into the flight, the shuttle breaks the sound barrier."
which is alot longer than what I thought it should take to reach the sound barrier.

*length=1:11

The duration of this video has been updated from unknown to 1:11 - length declared by ReverendTed.