Wavelet systems can be used as bases in quantum mechanical applications where localization and scale are both important. General quadrature formulas are developed for accurate evaluation of integrals involving compact support wavelet families, and their use is demonstrated in examples of spectral analysis and integrals over anharmonic potentials. In contrast to usual expectations for these uniformly spaced basis functions, it is shown that nonuniform spacings of sample points are readily allowed. Adaptive wavelet quadrature schemes are also presented for the purpose of meeting specific accuracy criteria without excessive oversampling.