It is well-known that the rational cohomology ring of a toric variety with orbifold singularities, say toric orbifolds, behaves similarly to the integral cohomology ring of smooth toric varieties. However, the information about the integral cohomology ring of a (underlying topological spaces of) toric orbifold or a singular toric variety in general is somewhat restrictive compared to the smooth case. In this talk, we consider toric surfaces, namely the toric varieties of complex dimensions 2, which have at worst orbifold singularities. The main result gives an explicit formula for the cup product structure with respect to a certain basis having a topological nature. This is a joint work with X. Fu and T. So.