In the framework of the discrete dipole approximation we develop a theoretical approach that allows the analysis of the role of multipole modes in the extinction and scattering spectra of arbitrary shaped nanoparticles. The main attention is given to the first multipoles including magnetic dipole and electric quadrupole moments. The role of magnetic quadrupole and electric octupole modes is also discussed. The method is applied to nonspherical Si nanoparticles with resonant multipole responses in the visible optical range, allowing a decomposition of single extinction (scattering) peaks into their constituent multipole contributions. It is shown by numerical simulations that it is possible to design silicon particles for which the electric dipole and magnetic dipole resonances are located at the same wavelength under certain propagation directions of incident light, providing new possibilities in metamaterial developments.