A method for analyzing and classifying two-dimensional pure point diffraction spectra (i.e. a set of Bragg peaks) of certain self-similar structures with scaling factor beta > 1, such as quasicrystals, is presented. The two-dimensional pure point diffraction spectrum Pi is viewed as a point set in the complex plane in which each point is assigned a positive number, its Bragg intensity. Then, by using a nested sequence of self-similar subsets called beta-lattices, we implement a multiresolution analysis of the spectrum Pi. This analysis yields a partition of Pi simultaneously in geometry, in scale and in intensity (the 'fingerprint' of the spectrum, not of the diffracting structure itself). The method is tested through numerical explorations of pure point diffraction spectra of various mathematical structures and also with the diffraction pattern of a realistic model of a quasicrystal.