  We study the gonality and canonical model of a rational unicuspidal curve C.
We are mainly interested in the case where C is non-Gorenstein. We classify
such curves via different notions of gonality, and by its canonical model C',
up to genus 6. We do it by means of more general families of curves of
arbitrary genus. Afterwards, we get a general formula for the dimension of the
space of hypersurfaces of a fixed degree containing C', which we apply to some
particular cases.
