  This note pursues the techniques of modified psi classes on the stack of
stable maps (cf. [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.
