The structure of a thermonuclear detonation wave can be solved accurately and, thus, may serve as a test bed for studying different approximations that are included in multidimensional hydrodynamical simulations of supernova. We present the structure of thermonuclear detonations for the equal mass fraction of C-12 and O-16 (CO) and for pure He-4 (He) over a wide range of upstream plasma conditions. The lists of isotopes we constructed allow us to determine the detonation speeds, as well as the final states for these detonations, with an uncertainty of the per cent level (obtained here for the first time). We provide our results with a numerical accuracy of similar to 0.1 per cent, which provides an efficient benchmark for future studies. We further show that CO detonations are pathological for all upstream density values, which differs from previous studies, which concluded that for low upstream densities CO detonations are of the Chapman-Jouget (CJ) type. We provide an approximate condition, independent of reaction rates, that allows to estimate whether arbitrary upstream values will support a detonation wave of the CJ type. Using this argument, we are able to show that CO detonations are pathological and to verify that He detonations are of the CJ type, as was previously claimed for He. Our analysis of the reactions that control the approach to nuclear statistical equilibrium, which determines the length-scale of this stage, reveals that at high densities, the reactions B-11+p 3(4)He plays a significant role, which was previously unknown.