  We analyze the fragmentation behavior of random clusters on the lattice under
a process where bonds between neighboring sites are successively broken.
Modeling such structures by configurations of a generalized Potts or
random-cluster model allows us to discuss a wide range of systems with fractal
properties including trees as well as dense clusters. We present exact results
for the densities of fragmenting edges and the distribution of fragment sizes
for critical clusters in two dimensions. Dynamical fragmentation with a size
cutoff leads to broad distributions of fragment sizes. The resulting power laws
are shown to encode characteristic fingerprints of the fragmented objects.
