We study the possibility for an isotropic elastic body to support forms of instability induced by shear stress states which are reminiscent of the planar Couette and the twisting Taylor- Couette patterns observed in the flow of viscous fluids. Here, we investigate the emergence of bifurcating periodic deformations for an infinitely long compressible elastic block confined between and attached to parallel plates which are subject to a relative shear displacement. We specialize our analysis by considering a generalized form of the Blatz-Ko strain energy function and show through numerical representative examples that planar Couette modes are always preferred with respect to the twisting Taylor-Couette modes. Finally, we describe some issues of such periodic bifurcating instabilities.