We review the linear stability analysis of the Navier-Stokes (NS) equation, and consider the Loose-Hess stability analysis of simulated atomic and molecular fluids under planar Couette flow. Traditional linear stability analysis of the NS equation fails to predict the critical Reynolds number where steady-state fluid flow becomes unstable. This failure arises because this analysis only treats perturbations to the steady-state flow which are single-eigenmode solutions of the linearized NS equation. By contrast, modern stability analysis of the NS equation, that considers perturbations which are superpositions of such modes, predicts that planar Couette flow can become unstable at Reynolds numbers less than that predicted by the traditional analysis. We show that the Loose-Hess stability theory, whose derivation resembles the traditional analysis of the NS equation, can accurately predict the critical shear rates where the flows of Lennard-Jones and simple dipolar fluids cease to be stable. Furthermore, noting the predictions made by the stability analyses of the NS equation, we describe the global stability of planar Couette flow simulated by molecular dynamics. (C) 2003 American Institute of Physics.