A comprehensive assessment is made of transient and asymptotic two-dimensional perturbation 
growth in compressible shear flow using unbounded constant shear and the Couette problem as 
examples. The unbounded shear flow example captures the essential dynamics of the rapid transient 
growth processes at high Mach numbers, while excitation by nonmodal mechanisms of nearly 
neutral modes supported by boundaries in the Couette problem is found to be important in sustaining 
high perturbation amplitude at long times. The optimal growth of two-dimensional perturbations in 
viscous high Mach number flows in both unbounded shear flow and the Couette problem is shown 
to greatly exceed the optimal growth obtained in incompressible flows at the same Reynolds 
number.