Distance squared functions on singular surfaces parameterized by smooth maps A-equivalent to Sₖ, Bₖ, Cₖ and F₄

We describe singularities of distance squared functions on singular surfaces in R³ parameterized by smooth map-germs A-equivalent to one of Sₖ, Bₖ, Cₖ and F₄ singularities in terms of extended geometric language via finite succession of blowing-ups. We investigate singularities of wave-fronts and caustics of such singular surfaces.