  We complete a full classification of non-degenerate traveling waves of scalar
balance laws from the point of view of spectral and nonlinear
stability/instability under (piecewise) smooth perturbations. A striking
feature of our analysis is the elucidation of the prominent role of
characteristic points in the determination of both the spectra of the
linearized operators and the phase dynamics involved in the nonlinear
large-time evolution. For a generic class of equations an upshot of our
analysis is a dramatic reduction from a tremendously wide variety of entropic
traveling waves to a relatively small range of stable entropic traveling waves.
