Context. Molecular hydrogen, H2, is the most abundant molecule in the Universe. Thanks to its widely spaced energy levels, it predominantly lights up in warm gas, T ≳ 102 K, such as shocked regions externally irradiated or not by interstellar UV photons, and it is one of the prime targets of James Webb Space Telescope (JWST) observations. These may include shocks from protostellar outflows, supernova remnants impinging on molecular clouds, all the way up to starburst galaxies and active galactic nuclei. Aims. Sophisticated shock models are able to simulate H2 emission from such shocked regions. We aim to explore H2 excitation using shock models, and to test over which parameter space distinct signatures are produced in H2 emission. Methods. We here present simulated H2 emission using the Paris-Durham shock code over an extensive grid of ~14 000 plane-parallel stationary shock models, a large subset of which are exposed to a semi-isotropic external UV radiation field. The grid samples six input parameters: the preshock density, shock velocity, transverse magnetic field strength, UV radiation field strength, the cosmic-ray-ionization rate, and the abundance of polycyclic aromatic hydrocarbons, PAHs. Physical quantities resulting from our self-consistent calculations, such as temperature, density, and width, have been extracted along with H2 integrated line intensities. These simulations and results are publicly available on the Interstellar Medium Services platform. Results. The strength of the transverse magnetic field, as quantified by the magnetic scaling factor, b, plays a key role in the excitation of H2. At low values of b (≲0.3, J-type shocks), H2 excitation is dominated by vibrationally excited lines; whereas, at higher values (b ≳ 1, C-type shocks), rotational lines dominate the spectrum for shocks with an external radiation field comparable to (or lower than) the solar neighborhood. Shocks with b ≫ 1 can potentially be spatially resolved with JWST for nearby objects. H2 is typically the dominant coolant at lower densities (≲104 cm-3); at higher densities, other molecules such as CO, OH, and H2O take over at velocities ≲20 km s-1 and atoms, for example, H, O, and S, dominate at higher velocities. Together, the velocity and density set the input kinetic energy flux. When this increases, the excitation and integrated intensity of H2 increases similarly. An external UV field mainly serves to increase the excitation, particularly for shocks where the input radiation energy is comparable to the input kinetic energy flux. These results provide an overview of the energetic reprocessing of input kinetic energy flux and the resulting H2 line emission.