  We investigate a class of stochastic fragmentation processes involving stable
and unstable fragments. We solve analytically for the fragment length density
and find that a generic algebraic divergence characterizes its small-size tail.
Furthermore, the entire range of acceptable values of decay exponent consistent
with the length conservation can be realized. We show that the stochastic
fragmentation process is non-self-averaging as moments exhibit significant
sample-to-sample fluctuations. Additionally, we find that the distributions of
the moments and of extremal characteristics possess an infinite set of
progressively weaker singularities.
