High-index dielectric or semiconductor nanoparticles support strong Mie-like geometrical resonances in the visible spectral range. We use 30 keV angle-resolved cathodoluminescence imaging spectroscopy to excite and detect these resonant modes in single silicon nanocylinders with diameters ranging from 60 - 350 nm. Resonances are observed with wavelengths in the range 400 - 700 nm, with quality factors in the range Q = 9 - 77, and show a strong redshift with increasing cylinder diameter. The photonic wavefunction of all modes is determined at deep-subwavelength resolution and shows good correspondence with numerical simulations. An analytical model is developed that describes the resonant Mie-like optical eigenmodes in the silicon cylinders using an effective index of a slab waveguide mode. It shows good overall agreement with the experimental results and enables qualification of all resonances with azimuthal (m = 0 - 4) and radial (q = 1 - 4) quantum numbers. The single resonant Si nanocylinders show characteristic angular radiation distributions in agreement with the modal symmetry.