This paper discusses wavelet thresholding in smoothing from non-equispaced, noisy data using so called second generation wavelets. We explain that a good numerical condition is an absolute requisite for successful thresholding. We examine the nature and origin of stability problems in second generation wavelet transforms, i.e. when applying a lifting scheme to non-equispaced data. The investigation concentrates on lifting with interpolating prediction, but the conclusions are extendible. The paper proposes two ways to stabilize the second generation wavelet transform. The first operates on the interval boundaries, while the second concentrates on the irregularity of the data points. Illustrations show that reconstruction from thresholded coefficients with this stabilized second generation wavelet transform leads to smooth and close fits.