In this paper we study the divisibility of the group $CH^2(X)$ of zero-cycles on a singular surface $X$ over a field $k$. Our results generalize known facts about the Chow group of a nonsingular surface. When $k$ is algebraically closed, we show that the subgroup of cycles of “degree”zero is divisible. When $k$ is the real numbers $R$, the non-divisible torsion of $CH^2(X)$ is detected by the topology of the stratified manifold $X({\mathbb R})$.