In this paper (which is essentially A. Swaminathan's undergraduate thesis), we introduce a curious infinite sequence of integers which together encode how much an abstract ICIS curve singularity germ is inflected. We explain how to compute this sequence (which we call the automatic degeneracies) for some of the simplest curve singularities, and we pose various questions about the sequence in general. For instance, one important lingering question is the eventual polynomiality of sequences of automatic degeneracies. It would also be interesting to know whether one could mimic the construction somehow to get analogous inflectionary invariants for ICIS singularities in higher dimensions.