We construct spline (finite element) spaces of variable approximation order and find necessary and sufficient conditions for pseudosmoothness of such splines. We study embedding of the spline spaces on embedded subdivisions and construct the corresponding wavelet decompositions. The constructions are based on the approximation relations defined on a cell subdivision of a differentiable manifold under the assumption that the multiplicity of the covering by supports of the coordinate functions is variable, which causes the variable approximation order. The spline spaces possess the adaptive approximation property. The notion of pseudosmoothness lead to new families of embedded spaces.