  In this paper, we study the linear stability properties of perturbations
around the homogeneous Couette flow for a 2D isentropic compressible fluid in
the domain $\mathbb{T}\times \mathbb{R}$. In the inviscid case there is a
generic Lyapunov type instability for the density and the irrotational
component of the velocity field. More precisely, we prove that their $L^2$ norm
grows as $t^{1/2}$ and this confirms previous observations in the physics
literature. Instead, the solenoidal component of the velocity field experience
inviscid damping, meaning that it decays to zero even in the absence of
viscosity. For a viscous compressible fluid, we show that the perturbations may
have a transient growth of order $\nu^{-1/6}$ (with $\nu^{-1}$ being
proportional to the Reynolds number) on a time-scale $\nu^{-1/3}$, after which
it decays exponentially fast. This phenomenon is also called enhanced
dissipation and our result appears to be the first to detect this mechanism for
a compressible fluid, where an exponential decay for the density is not a
priori trivial given the absence of dissipation in the continuity equation.
