Summary: We introduce a simple model of two conservation laws which is strictly hyperbolic except for a degenerate parabolic line in the state space. Besides classical shock waves, it also exhibits overcompressive, marginal overcompressive, and marginal undercompressive shock waves. Our purpose is to study the behavior of the corresponding viscous waves, in particular the manner in which these waves are stable. There are several basic differences between classical shock waves and other types of shock waves. A perturbation of an overcompressive shock wave gives rise to a new wave. Monotone marginal overcompressive waves behave distinctly from the nonmonotone ones. Analytical techniques used in our study include characteristic-energy and weighted-energy methods, and nonlinear superposition through time-invariants. Although we carry out our analysis for a simple model, the general phenomena would hold for overcompressive waves which occur in other physical models.