Turbulent dissipation is a central issue in the star and galaxy formation process. Its fundamental property of space-time intermittency, well characterised in incompressible laboratory experiments, remains elusive in cosmic turbulence. Progress requires the combination of state-of-the-art modelling, numerical simulations and observations. The power of such a combination is illustrated here where the statistical method intended to locate extrema of velocity shears in a turbulent field is applied to numerical simulations of compressible magneto-hydrodynamical (MHD) turbulence dedicated to dissipation scales and to a nearby turbulent diffuse molecular cloud. We demonstrate that short-spacing increments of observables can detect strongly dissipative structures. In our simulations, we compute structure functions of various synthetic observables and show that they verify Extended Self-Similarity. This allows to compute their intermittency exponents and we show how they could help constraining some properties of the underlying three-dimensional turbulence. In observations of a turbulent cloud close to the Sun in our Galaxy, a remarkable coherent structure of velocity shear extremum is disclosed . At the location of the largest velocity shear, this coherent structure is found to foster 10 × more carbon monoxide molecules than standard diffuse molecular gas, an enrichment supported by models of non-equilibrium warm chemistry triggered by turbulent dissipation. The power of the combination between modelling and observations is also illustrated by observations of the CH+ cation that provide unique quantitative informations on the kinetic energy trail in the massive, multi-phase and turbulent circum-galactic medium of a galaxy group at redshift z = 2.8.