A theoretical approach, allowing analyzing the role of multipole modes in the extinction and scattering spectra of arbitrary shaped nanoparticles, is developed in the framework of the discrete dipole approximation. The proposed method can be used to control separately the positions of different multipole resonances as a function of nanoparticle sizes, shapes and irradiation conditions. The main attention is given to the first multipole modes including magnetic dipole and electric quadrupole moments. The magnetic quadrupole and electric octupole modes can also be involved in the consideration. The method is applied to nonspherical Si nanoparticles with multipole responses in the visible optical range, allowing a decomposition of single extinction (scattering) peaks into their constituting multipole contributions. The unique property of Si nanoparticles to support magnetic optical response opens new ways for the construction of novel nanooptical elements and can be particularly important for solving the problem of metamaterials with magnetic properties in the visible spectral range.