  We study the linear stability of holographic homogeneous solids (HHS) at
finite temperature and in presence of a background shear strain by means of a
large scale quasi-normal mode analysis which extends beyond the hydrodynamic
limit. We find that mechanical instability can arise either as a result of a
complex speed of sound -- gradient instability -- or of a negative diffusion
constant. Surprisingly, the simplest HHS models are linearly stable for
arbitrarily large values of the background strain. For more complex HHS, the
onset of the diffusive instability always precedes that of the gradient
instability, which becomes the dominant destabilizing process only above a
critical value of the background shear strain. Finally, we observe that the
critical strains for the two instabilities approach each other at low
temperatures. We conclude by presenting a phase diagram for HHS as a function
of temperature and background shear strain which shows interesting similarities
with the physics of superfluids in presence of background superfluid velocity.
