Context. An essential facet of turbulence is the space-time intermittency of the cascade of energy that leads to coherent structures of high dissipation. Aims. In this work, we aim to systematically investigate the physical nature of the intense dissipation regions in decaying isothermal magnetohydrodynamical (MHD) turbulence. Methods. We probed the turbulent dissipation with grid-based simulations of compressible isothermal decaying MHD turbulence. We took unprecedented care in resolving and controlling dissipation: we designed methods to locally recover the dissipation due to the numerical scheme. We locally investigated the geometry of the gradients of the fluid state variables. We developed a method to assess the physical nature of the largest gradients in simulations and to estimate their travelling velocity. Finally, we investigated their statistics.Results. We find that intense dissipation regions mainly correspond to sheets; locally, density, velocity, and magnetic fields vary primarily in one direction. We identify these highly dissipative regions as fast and slow shocks or Alfvén discontinuities (Parker sheets or rotational discontinuities). On these structures, we find the main deviation from a 1D planar steady-state is mass loss in the plane of the structure. We investigated the effect of initial conditions, which yield different imprints at an early time on the relative distributions among these four categories. However, these differences fade out after about one turnover time, at which point they become dominated by weakly compressible Alfvén discontinuities. We show that the magnetic Prandtl number has little influence on the statistics of these discontinuities, but it controls the ohmic versus viscous heating rates within them. Finally, we find that the entrance characteristics of the structures (such as entrance velocity and magnetic pressure) are strongly correlated. Conclusions. These new methods allow us to consider developed compressible turbulence as a statistical collection of intense dissipation structures. This can be used to post-process 3D turbulence with detailed 1D models apt for comparison with observations. It could also be useful as a framework to formulate new dynamical properties of turbulence.