We discuss the construction of three-point finite difference approximations and their convergence for the class of singular two-point boundary value problems: (xαy′)′=f(x,y), y(0)=A, y(1)=B, 0<α<1. We first establish a certain identity, based on general (non-uniform) mesh, from which various methods can be derived. To obtain a method having order two for all α∈(0,1), we investigate three possibilities. By employing an appropriate non-uniform mesh over [0,1], we obtain a methodM1 based on just one evaluation off. For uniform mesh we obtain two methodsM2 andM3 each based on three evaluations off. For α=0,M1 andM2 both reduce to the classical second-order method based on one evaluation off. These three methods are investigated, theirO(h2)-convergence established and illustrated by numerical examples.