Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)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 the canonical model of a rational monomial curve with just one singular point lies on a surface scroll if and only if the gonality of the curve is at most 3, and that it lies on a threefold scroll if and only if the gonality is at most 4.275FAPESP [07481-1/2012, 19676-7/2014]CNPq [307978/2012-5]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)