The analytical methods of isolation and description of the resonance components of partial modes in scattering problems of acoustic, elastic, and electromagnetic waves are set forth. The results of these methods' applications are presented and their effectiveness is discussed. For some particular cases the exactness of the results obtained can be estimated as well. The following methods are considered: gradient of the phase, utilization of impedance type intermediate backgrounds, Argand diagram method, and pole evaluation on the frequency complex plane for a fixed mode order. The methods can be even used for rather complicated situations: for very narrow and rather broad resonances, near the point of intersection of the dispersion curves of phase velocities, and for very close situated resonances. The illustrative examples are given for the problems of scattering of bodies of spherical and cylindrical shape. The resonances are isolated for waves of different physical nature: diffracted (Franz type), normal (Lamb type), shear, Rayleigh type, and whispering gallery. With the properly computed modal resonances one can immediately obtain the dispersion curves of the phase and group velocities of every wave, and construct the acoustic spectrogram of the scatterer. Jointly with the well-known experimental methods, the described ones form the basis for ultrasonic spectroscopy.