I am looking for a reference that describes what are the possible singularities of a curve $C=C_1 \cup C_2$ which is the union of two smooth irreducible curves $C_1,\,C_2$ on a smooth surface.

Doing some computations, my guess is that these singularities are always of type $A_{2m-1}$ ($m\in \mathbb{N}^*$), but I would like to see a clear proof.