Singular splines are required for approximation of functions with point singularities. This paper develops a concept ofgeneralized smoothness in the case of the mentioned splines. Generalized smoothness is defined as equality of values for two functionals with local supports. The splines are constructed by the suitable approximation relations. The last ones are obtained by application of the concept to the right part of the relations (i.e. to generating vector function). Sufficient conditions of coordinate spline existence are established. The generalized smoothness of the coordinate splines has as a consequence of embedded spline spaces constructed on embedded grids.