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Nanodiamonds with color centers have found applications in a variety of fields ranging from biophotonics to quantum optics due to the highly stable non-bleaching luminescence in the visible spectrum. Recently, Mie resonances in high-permittivity nanoparticles have attracted significant research attention due to their ability to enhance electromagnetic fields inside the material and control the propagation of light on the nanoscale. The high refractive index of diamond (n=2.4 in the visible) allows for the excitation of Mie resonances in subwavelength diamond nanoparticles that can be used to increase the efficiency of the operation of color centers in the nanodiamonds. We report on an experimental study of Mie resonances in single nanodiamonds. The particles with sizes from 300 to 400 nm were made by chemical vapor deposition and characterized using scanning electron microscopy. Scattering spectra of individual nanodiamonds with predetermined size and shape were measured using single-particle dark-field scattering spectroscopy. In the spectra, pronounced peaks corresponding to the magnetic dipole and quadrupole resonances are revealed. These peaks are shown to experience a redshift with increasing particle size. In the case of a strongly anisotropic particle, the scattering is shown to be modified depending on the polarization of light. The experimental results are validated by comparison with both analytical and numerical calculations. The position and quality of the resonances obtained experimentally for the real-shape particles are in agreement with the results calculated for spherical particles. In the case of an anisotropic particle, the spheroid approximation gives appropriate results. Finally, we discuss the applications of the studied resonances and numerically demonstrate an almost two-order enhancement of the electric Purcell factor in the magnetic quadrupole resonance compared with that in the case of a non-resonant nanoparticle, making Mie-resonant nanodiamonds a strong case for nanophotonics.