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 attempt to investigate systematically the physical nature of the intense dissipation regions in decaying isothermal magnetohydrodynamical (MHD) turbulence. Methods. We probe the turbulent dissipation with grid based simulations of compressible isothermal decaying MHD turbulence. We take unprecedented care at resolving and controlling dissipation: we design methods to locally recover the dissipation due to the numerical scheme. We locally investigate the geometry of the gradients of the fluid state variables. We develop a method to assess the physical nature of the largest gradients in simulations and to estimate their travelling velocity. Finally we investigate their statistics. Results. We find that intense dissipation regions mainly correspond to sheets: locally, density, velocity and magnetic fields vary primarily across one direction. We identify these highly dissipative regions as fast/slow shocks or Alfvén discontinuities (Parker sheets or rotational discontinuities). On these structures, we find the main deviation from 1D planar steady-state is mass loss in the plane of the structure. We investigate the effect of initial conditions which yield different imprints at early time on the relative distributions between these four categories. However, these differences fade out after about one turnover time, when 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 vs viscous heating rates within them. Finally, we find the entrance characteristics of the structures (such as entrance velocity and magnetic pressure) are strongly correlated. Conclusions. These new methods allow 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 reveal useful as a framework to formulate new dynamical properties of turbulence.