The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.
"Characterizing singularities of a surface in Lie sphere geometry." Hokkaido Math. J. 48 (2) 281 - 308, June 2019. https://doi.org/10.14492/hokmj/1562810509