This note pursues the techniques of [Graber-Kock-Pandharipande] to give concise solutions to the characteristic number problem of rational curves in P-2 or P-1 x P-1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.