In this paper we use the lifting scheme to construct biorthogonal spline wavelet bases on regularly refined non-uniform grids. The wavelets have at least one vanishing moment and on each resolution level they form an L2 Riesz basis. Furthermore we are interested in determining the exact range of Sobolev exponents for which the complete multilevel system forms a Riesz basis. Hereto we need to examine the smoothness of the dual scaling functions which involves investigation of the spectral properties of the associated transition operator. We provide several examples and discuss their stability. Furthermore we also give a strategy to construct biorthogonal spline wavelets on uniform grids.