For incompressible flows, the flow field can exhibit a checkboard patterned instability. Can this happen also for compressible flows? Do you know literature for this?

I shouldn't think so. Using the FEM, the velocities are usually good in the presence of the "instability" unless you really screw up your code. The checkerboard should not really be a concern. There is a confusion between the function which projects out the non-solenoidal part of the flow (and which must satisfy the LBB condition) and the pressure (which need not satisfy the LBB condition) because they obey similar equations. The projection function is non-physical and if it exhibits checkerboard, so what?, as long as the velocity is OK. Properly computed from the flow field, the pressure will not checkerboard. The checkerboarding in the projection function arises from the fact that on meshes topologically equivalent to a checkerboard, the continuity constraint must be satisfied separately on the red and on the black squares.