We study viscous shock waves that are associated with a simple mode (lambda, r) of a system u(t) + f(u)x= u(xx) of conservation laws and that connect states on either side of an `inflection' hypersurface Sigma in state space at whose points r center dot del lambda=0 and (r center dot del)(2)lambda not equal 0. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law u(t)+(u(3))(x)=u(xx), occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves. (C) 2014 Elsevier Inc. All rights reserved.