- Stellar discs of highly flattened giant galaxies, including that of the Milky Way, are studied by linear theory to determine the stability of such discs against small-amplitude gravity perturbations. In order to understand the physics of the problem better, the simplest theoretical model is applied. That is, the local disc is studied by employing the method of particle orbit theory. In this purely Lagrangian method, an approximate solution of the Newtonian equations of the motion of stars is obtained using a general technique based upon the perturbation method. In the second order of Lindblad’s epicyclic theory, expressions are found for the unperturbed motions of stars in a stationary system with an axially symmetric mass distribution. Then, expressions are found for the perturbed motions of stars when the small non-axisymmetric gravity perturbation is additionally taken into account. The perturbed terms are obtained as second-order oscillations. To describe the ordered behaviour of a medium near its quasi-equilibrium state, these equations for the trajectories of stars are used to obtain the dispersion relation that connects the frequency of excited collective oscillations with the wavenumber throughout the disc, including resonant regions. Using the dispersion relation, a new class of gradient microinstabilities of a non-uniformly rotating disc inherent in an inhomogeneous system is discussed. The Landau mechanism of excitation of spiral density waves works at the corotation resonance between stars and hydrodynamically (Jeans) stable perturbations (e.g. those produced by a bar-like structure, a spontaneous perturbation and/or a companion galaxy). A physical aetiology of the gradient microinstabilities of collisionless stellar discs is explained. Such instabilities can develop only if the inhomogeneous and non-uniformly rotating disc of stars is Jeans-stable. Certain astronomical implications of the theory for actual galaxies are explored as well. In particular, the development of these instabilities of a stellar disc can result directly in the formation of different observable structural features, e.g. spiral arms and collisionless dynamical relaxation of the system on the Hubble time-scale.