This study is of a review nature. In section 1 is described the method of investigation of the stability of stationary oscillations of not too large amplitude (the ratio of the hydrodynamic wave velocity to phase velocity serves as a small parameter). Instabilities arise because of the presence of a positive feedback between small sinusoidal oscillations propagating in the "background" of the finite-amplitude wave. Instabilities recall quasi-particle decays: frequencies and wave vectors of developing oscillations are related to the frequency and wave vector of the initial wave by relations typical of the laws of energy and momentum conservation. A characteristic peculiarity of growth rates of developing oscillations is their proportionality to the value of the initial wave amplitude.A concrete example of finding the growth rate of small perturbations is discussed; the stability of a sinusoidal Alfvén wave of small amplitude (section 2A) is investigated. Instability of such a wave is related to compressibility of the medium (for incompressible media instability disappears).For large-amplitude oscillations there is no general method for stability investigation. However even here a positive feedback between small oscillations can appear. For this reason an Alfvén wave is unstable (in a compressible medium!) if it has an arbitrary amplitude with a sawtooth profile for the magnetic field lines (the chosen profile of the magnetic field lines enables one to solve the problem precisely (section 2B)).In the concluding portion of the review (sections 3 and 4) the results of investigation of the stability of other oscillation modes are given, such as ion-acoustic oscillations of an non-isothermal plasma, electron Langmuir oscillations, magneto-acoustic oscillations, etc.