We present a new approach of nonparametric regression with wavelets if the design is stochastic. In contrast to existing approaches we use a new construction of a design-adapted wavelet basis which is constructed given the random regressors. We first treat the case of using orthogonal design-adapted Haar wavelets for regression with (non-Gaussian) i.i.d. errors. We derive results on the near-optimal rate of convergence of the minimax L2-risk of non-linear threshold estimators over a certain function class which parallel those of the classical case of fixed equidistant design. In a second part of the talk, we present both on a theoretical and practical ground how the Haar basis can be improved using the so-called Lifting Scheme. This leads to the construction of smoother biorthogonal bases which are still adapted to the stochastic design. We propose a thresholding scheme and show on simulated and real datasets the performance of the method.