  Let $C$ be an integral and projective curve; and let $C'$ be its canonical
model. We study the relation between the gonality of $C$ and the dimension of a
rational normal scroll $S$ where $C'$ can lie on. We are mainly interested in
the case where $C$ is singular, or even non-Gorenstein, in which case
$C'\not\cong C$. We first analyze some properties of an inclusion $C'\subset S$
when it is induced by a pencil on $C$. Afterwards, in an opposite direction, we
assume $C'$ lies on a certain scroll, and check some properties $C$ may
satisfy, such as gonality and the kind of its singularities. At the end, we
prove that a rational monomial curve $C$ has gonality $d$ if and only if $C'$
lies on a $(d-1)$-fold scroll.
