The Orr–Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.