  Let $C$ be an integral and projective curve whose canonical model $C'$ lies
on a rational normal scroll $S$ of dimension $n$. We mainly study some
properties on $C$, such as gonality and the kind of singularities, in the case
where $n=2$ and $C$ is non-Gorenstein, and in the case where $n=3$, the scroll
$S$ is smooth, and $C'$ is a local complete intersection inside $S$. We also
prove that a rational monomial curve with just one singular point lies on a
surface scroll if and only if its gonality is at most $3$, and that it lies on
a threefold scroll if and only if its gonality is at most $4$.
