We report experimental studies of spin-wave excitations in individual 22-nm-thick Ni80Fe20 nanotubes with diameters of about 150 nm. We apply Brillouin light-scattering spectroscopy under microwave irradiation, and we resolve sets of discrete resonances in the center of nanotubes ranging from 2.5 to 12.5 GHz. Comparing to a recent theoretical work and micromagnetic simulations, we identify different characteristic eigenmodes depending on the axial, mixed, or vortex configuration. The mixed and vortex states give rise to modes with helical phase profiles substantiating an unusual nature of confined modes attributed to nonreciprocal spin waves. Our findings provide microscopic insight into realistic tubular spin-wave nanocavities and magnetochiral effects for three-dimensional nanomagnonics.