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 the wavelet decomposition of spline spaces it is important that the property of the embedding of these spaces is associated with embedding grids. The purpose of this paper is to consider ways of constructing spaces of splines with a predefined singularity and obtain their wavelet decomposition. Here the concept of generalized smoothness is used, within which the mentioned singularity is generalized smooth. This approach leads to the construction of a system of embedded spaces on embedded grids. A spline-wavelet decomposition of mentioned spaces is presented. Reconstruction formulas are done.