  In many-body systems with quenched disorder, dynamical observables can be
singular not only at the critical point, but in an extended region of the
paramagnetic phase as well. These Griffiths singularities are due to rare
regions, which are locally in the ordered phase and contribute to a large
susceptibility. Here, we study the geometrical properties of rare regions in
the transverse Ising model with dilution or with random couplings and
transverse fields. In diluted models, the rare regions are percolation
clusters, while in random models the ground state consists of a set of spin
clusters, which are calculated by the strong disorder renormalization method.
We consider the so called energy cluster, which has the smallest excitation
energy and calculate its mass and linear extension in one-, two- and
three-dimensions. Both average quantities are found to grow logarithmically
with the linear size of the sample. Consequently, the rare regions are not
compact: for the diluted model they are isotropic and tree-like, while for the
random model they are quasi-one-dimensional.
