The spectrum of linear modes governing the multidimensional instabilities of gaseous detonations is revisited by combining a numerical analysis with new analytical results. In view of recent develop`ments in nonlinear analyses for describing the cellular structure of weakly unstable detonation fronts, particular attention is paid to the neighbourhood of the instability threshold. A first objective is to check the validity domain of the analytical results and to investigate to what extent they are useful when approaching the self-sustained regime (Chapman–Jouguet conditions). A second objective is to study how the multidimensional instabilities are influenced by multiple-step chemistry. The roles of the induction period and of the stiffness of the exothermic runaway will be investigated separately.