We study analytically the single-trajectory spectral density (STSD) of an active Brownian motion (BM) as exhibited, for example, by the dynamics of a chemically-active Janus colloid. We evaluate the standardly-defined spectral density, ie the STSD averaged over a statistical ensemble of trajectories in the limit of an infinitely long observation time T, and also go beyond the standard analysis by considering the coefficient of variation γ of the distribution of the STSD. Moreover, we analyse the finite-T behaviour of the STSD and γ, determine the cross-correlations between spatial components of the STSD, and address the effects of translational diffusion on the functional forms of spectral densities. The exact expressions that we obtain unveil many distinctive features of active BM compared to its passive counterpart, which allow to distinguish between these two classes based solely on the spectral content of individual trajectories.