  The Kitaev model is a fascinating example of an exactly solvable model
displaying a spin-liquid ground state in two dimensions. However, deviations
from the original Kitaev model are expected to appear in real materials. In
this Letter, we investigate the fate of Kitaev's spin liquid in the presence of
disorder -- bond defects or vacancies -- for an extended version of the model.
Considering static flux backgrounds, we observe a power-law divergence in the
low-energy limit of the density of states with a nonuniversal exponent. We link
this power-law distribution of energy scales to weakly coupled droplets inside
the bulk, in an uncanny similarity to the Griffiths phase often present in the
vicinity of disordered quantum phase transitions. If time-reversal symmetry is
broken, we find that power-law singularities are tied to the destruction of the
topological phase of the Kitaev model in the presence of bond disorder alone.
However, there is a transition from this topologically trivial phase with
power-law singularities to a topologically nontrivial one for weak to moderate
site dilution. Therefore, diluted Kitaev materials are potential candidates to
host Kitaev's chiral spin-liquid phase.
