We found that small perturbations of the optical vortex core in Laguerre-Gaussian (LG) beams generate a fine structure of the Hermite-Gaussian (HG) mode spectrum in the form of weak variations of amplitudes and phases of the HG modes. We developed and implemented the intensity moments technique for measuring the HG mode spectra. We also theoretically justified and experimentally implemented a technique for measuring the topological charge of the LG beams with an arbitrary number of ring dislocations. Theoretical discussion and experimental study are accompanied by examples of estimating the orbital angular momentum and the topological charge of perturbed LG beams as well as the algorithm for plotting the HG mode spectra. (C) 2020 Optical Society of America