Radiation and diffusion are inherent phenomena in Nature that anyone of us already witnessed in the form of, e.g., the propagation of gravity waves at the surface of water or the thermal conduction in solid structures. A certain emission of energy is evidently associated with the presence of damping in all sorts of dynamical systems, from the simplest spring-mass configurations to the more complex stellar structures. For instance, as historically predicted for the model of a rotating self-gravitating mass of fluid, presence of dissipation in the form of viscosity may lead to the onset of a secular instability, in the form of oscillatory motions. This discovery is nowadays well-accepted as the first illustration of the so-called dissipation-induced instability, a particular sort of instability arising in the presence of damping. A similar effect can be encountered in the context of the emission of waves carrying modes of opposite energy sign and yields the analogue radiation-induced instability. We propose hereafter along this thesis to present a selection of problems involving radiative or diffusive mechanisms and to carry out an exhaustive linear stability analysis on them. Respectively, the different chapters are ordered as referring to the stability of the Maclaurin spheroids, the lenticular vortices, the rotating magnetohydrodynamics flows and the fluid-structure interactions of an elastic membrane with a uniform potential flow. A particular attention will be addressed to systems subject to two simultaneous dissipative mechanisms, in order to estimate the predominance of the damped mode of importance. By means of various analytical treatments, as well as supporting numerical methods, we manage to solve the different boundary value problems associated with each system and thus establish new stability criteria in the spaces of parameters. The developed methods remain general in their formulation and are not solely designed to only solve the problems of interest. As a matter of fact, one can easily adapt our approach to a broad range of applications.