One of the approaches to the problem of approximating functions with a singularity is the creation of an approximating apparatus based on splines with the same feature. For spline-wavelet decomposition of spline spaces it is important that the property of the embedding of these spaces is associated with the embedding grids. In this paper the approximation relations are considered to determine coordinate splines with a predefined singularity. The concept of generalized smoothness is introduced. It allows us to consider functions with singularity as generalized smooth functions. Corresponding approximation relations are constructed. The existence and uniqueness of coordinate splines with the mentioned feature are established. The linear shells of the coordinate splines are spaces with the property of embedding on embedding grids.