  In this article we study rational curves with a unique unibranch genus-$g$
singularity, which is of {\it $\ka$-hyperelliptic} type in the sense of
\cite{To}; we focus on the cases $\ka=0$ and $\ka=1$, in which the semigroup
associated to the singularity is of (sub)maximal weight. We obtain a partial
classification of these curves according to the linear series they support, the
scrolls on which they lie, and their gonality.
