We provide a characterization of point defects in droplets of cholesteric liquid crystal, using a combination of experiment, simulation, and theoretical analysis. These droplets display a range of structures including realizations of defects with high topological charge and arrangements of multiple defects in “topological molecules”. We show that there are certain defects that are incompatible with a uniform sense of chiral twisting for topological reasons. Furthermore, those defects that are compatible with twist of a single handedness are shown to have the structure of the gradient field of an isolated critical point and, hence, are described by singularity theory. We show that the mathematical tools of singularity theory reproduce, with excellent agreement, the experimental observations of high charge defects and topological molecules. Our results have implications beyond liquid crystal droplets in characterizing chiral materials and their topology in general.