  We study parameter spaces of linear series on projective curves in the
presence of unibranch singularities, i.e. {\it cusps}; and to do so, we
stratify cusps according to value semigroup. We show that {\it generalized
Severi varieties} of maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images
of fixed degree and arithmetic genus are often {\it reducible} whenever $n \geq
3$. We also prove that the Severi variety of degree-$d$ maps with a
hyperelliptic cusp of delta-invariant $g \ll d$ is of codimension at least
$(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1
\rightarrow \mathbb{P}^n$; and that for small $g$, the bound is exact, and the
corresponding space of maps is the disjoint union of unirational strata.
Finally, we conjecture a generalization for unicuspidal rational curves
associated to an {\it arbitrary} value semigroup.
