  We fill the two main remaining gaps in the full classification of
non-degenerate planar traveling waves of scalar balance laws from the point of
view of spectral and nonlinear stability/instability under smooth
perturbations. On one hand we investigate the impact on the classification of
allowing to restrict perturbations by further localization constraints, the
so-called convective stability analysis. On the other hand, in the
multidimensional case we extend the existing analysis to general perturbations,
relaxing the artificial restriction, assumed so far, that initial perturbations
are supported away from characteristic points. A striking feature of the latter
analysis is that it involves infinite-dimensional families of genuinely
multidimensional traveling waves that live near the perturbed planar traveling
wave.
