Hadronic supercriticalities are radiative instabilities that appear when large amounts of energy are stored in relativistic protons. When the proton energy density exceeds some critical value, a runaway process is initiated resulting in the explosive transfer of the proton energy into electron-positron pairs and radiation and the increase of the photon-to-proton efficiency. We perform a comprehensive study of the parameter space by investigating the onset of hadronic supercriticalities for a wide range of source parameters (i.e., magnetic field strengths of 1 G−100 kG and radii of 10 11 cm−10 16 cm) and maximum proton Lorentz factors (10 3 − 10 9 ). We show that supercriticalities are possible for the whole range of source parameters related to compact astrophysical sources, like gamma-rays bursts, cores and jets of active galactic nuclei. We also provide an in-depth look at the physical mechanisms of hadronic supercriticalities and show that magnetized relativistic plasmas are excellent examples of non-linear dynamical systems in high-energy astrophysics.