  We describe singularities of distance squared functions on singular surfaces
in $\mathbb{R}^3$ parameterized by smooth map-germs $\mathcal{A}$-equivalent to
one of $S_k$, $B_k$, $C_k$ and $F_4$ 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.
