We present a shear instability, which can be triggered in compressible fluids with density-dependent viscosity at shear rates above critical. The instability mechanism is generic: It is based on density-dependent viscosity, compressibility, as well as flow two-(three-)dimensionality that provides coupling between streamwise and transversal velocity components and density variations. The only factor stabilizing the instability is fluid elasticity. The corresponding eigenvalue problem for a plane Couette flow is solved analytically in the limiting cases of large and small wave numbers.