  Let $C$ be a curve of genus at least three defined over a number field, and
let $r$ be the rank of the rational points of its Jacobian. Under mild
hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll
bound the number of rational points on $C$ by a constant that depends only on
its genus. Yet one expects an even stronger bound that depends favorably on
$r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper,
Stoll established such a "rank-favorable" bound for hyperelliptic curves using
Chabauty's method. In the present work we extend Stoll's results to
superelliptic curves, noting in the process some differences that ought to
inform uniformity conjectures for general curves. Our results have stark
implications for bounding numbers of rational points, since $r$ is expected to
be small for "most" curves.
