We compute the spectral distortions of the cosmic microwave background (CMB) arising during the epoch of cosmological hydrogen recombination within the standard cosmological (concordance) model for frequencies in the range 1-3500 GHz. We follow the evolution of the populations of the hydrogen levels including states up to principle quantum number n = 30 in the redshift range 500 <= z <= 3500. All angular momentum substates are treated individually, resulting in a total number of 465 hydrogen levels. The evolution of the matter temperature and the fraction of electrons coming from HeII are also included. We present a detailed discussion of the distortions arising from the main dipolar transitions, for example Lyman and Balmer series, as well as the emission due to the two-photon decay of the hydrogen 2s level. Furthermore, we investigate the robusteness of the results against changes in the number of shells considered. The resulting spectral distortions have a characteristic oscillatory behaviour, which might allow experimentalists to separate them from other backgrounds. The relative distortion of the spectrum exceeds a value of 10-7 at wavelengths longer than 21 cm. Our results also show the importance of detailed follow-up of the angular momentum substates, and their effect on the amplitude of the lines. The effect on the residual electron fraction is only moderate, and mainly occurs at low redshifts. The CMB angular power spectrum is changed by less than 1 per cent. Finally, our computations show that if the primordial radiation field is described by a pure blackbody, then there is no significant emission from any hydrogen transition at redshifts greater than z ~ 2000. This is in contrast to some earlier works, where the existence of a `pre-recombination' peak was claimed.