  We study the spectral stability of roll-wave solutions of the viscous St.
Venant equations modeling inclined shallow-water flow, both at onset in the
small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general
values of the Froude number $F$, including the limit $F\to +\infty$. In the
former, $F\to 2^+$, limit, the shallow water equations are formally
approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that
is a singular perturbation of the standard Korteweg de Vries (KdV) equation
modeling horizontal shallow water flow. Our main analytical result is to
rigorously validate this formal limit, showing that stability as $F\to 2^+$ is
equivalent to stability of the corresponding KdV-KS waves in the KdV limit.
Together with recent results obtained for KdV-KS by
Johnson--Noble--Rodrigues--Zumbrun and Barker, this gives not only the first
rigorous verification of stability for any single viscous St. Venant roll wave,
but a complete classification of stability in the weakly unstable limit. In the
remainder of the paper, we investigate numerically and analytically the
evolution of the stability diagram as Froude number increases to infinity.
Notably, we find transition at around $F=2.3$ from weakly unstable to
different, large-$F$ behavior, with stability determined by simple power law
relations. The latter stability criteria are potentially useful in hydraulic
engineering applications, for which typically $2.5\leq F\leq 6.0$.
