We prove that the variety of (parameterizations of) rational curves of sufficiently large fixed degree d in P^n with a single hyperelliptic cusp of delta-invariant g is always of codimension at least (n−1)g inside the space of degree-d holomorphic maps P^1→P^n; and that when g is small, this bound is exact and the corresponding space of maps is paved by unirational strata indexed by fixed ramification profiles. We give a conjectural generalization of this picture for rational curves with cusps of arbitrary value semigroup S, and provide evidence for this conjecture whenever S is a γ-hyperelliptic semigroup of either minimal or maximal weight. Finally, we produce infinitely many new examples of reducible Severi varieties M^n_{d,g} of holomorphic maps P^1→P^n with images of degree d and arithmetic genus g, for every value of n>2.