Let C be an irreducible projective plane curve defined over an algebraically closed field k. A point P on the plane is called a Galois point if the projection with center P induces a Galois extension of function fields, i.e., the normalization of C is a Galois covering of the projective line. Then, several questions arise, the standard ones are: (1) How is the structures of C and the Galois group when there exists a Galois point. (2) How is the distribution of Galois points. If the characteristic of k is zero and C is smooth, then they are simple. However, if not so, the question are rather difficult. We study them in detail in the case where C has a singular point and has a genus zero or one.