What does the label of stability: 0 mean on an MSDS sheet?
According to the Wikipedia site for the NFPA 704   identification system, a Stability rating of 0 means:  


  Normally stable, even under fire exposure conditions, and is not
  reactive with water.


The low concentration of the bleach, in other words it's mostly water, is the reason that it fits the "0" category.

Note that the bleaching component of the bleach solution is sodium hypochlorite (see your MSDS for bleach).  Pure solid sodium hypochlorite has a NFPA reactivity rating of "1".