The total spectrum of acoustic-gravity waves (AGWs), which can exist in an infinite isothermal atmosphere, is analyzed. The main attention in the work was paid to those areas of the spectrum that are forbidden for freely propagating waves. The term “forbidden” spectral region is conditional, since in this region there may exist waves of other types, which differ in the way of propagation and the properties from the freely propagating AGWs. In the spectrum of acoustic-gravity waves, it was studied different types of forbidden zones, which follow from the analysis of the dispersion equation obtained in the model of the infinite isothermal atmosphere. A classification of the types of forbidden areas in the AGW spectrum is proposed. The structure and the localization of the forbidden zones relative to the regions of freely propagating waves, as well as special points in the forbidden zones of the AGW spectrum, were studied on the corresponding spectral diagrams. In the forbidden region of type I, which separates the acoustic and gravitational regions of freely propagating AGWs, horizontal waves with a purely imaginary value of the vertical wave number can exist. The possibility of the existence of special acoustic-gravity modes in the AGW spectrum, for which one of the perturbed quantities is zero, was investigated, and it was shown that they can exist only in the forbidden spectral zone of type I. We also analyzed the forbidden zone in which vertical acoustic-gravity waves with a purely imaginary value of the horizontal wave number can exist. The paper also considered the spectral region in which the existence of acoustic-gravity waves is impossible, but atmospheric oscillations may be realized. The properties of wave solutions in various type forbidden spectral regions, including the peculiarities of polarization ratios, are also analyzed. The theoretical analysis of the forbidden spectral areas of the AGWs can be used for the experimental search of new types of wave solutions in the atmosphere.