Given an algebraic curve, one can ask the question, "How many inflection points of a particular type does it have?" The standard techniques for answering such enumerative questions break down when the curves under consideration are singular. In this talk, we discuss a new method to count inflection points on curves with l.c.i. singularities, and we explain how to use our method to answer various interesting enumerative questions, such as: (1) How many hyperflexes and septactic points are there in a general pencil of plane curves of a fixed degree? and (2) In a family of curves acquiring an isolated l.c.i. singularity of a given type, how many inflection points of a given type limit to the singularity? As we shall demonstrate, the answer to the second question is an invariant of the analytic isomorphism class of the singularity and can be computed algorithmically in many interesting examples. This work is joint with Anand Patel.