We use hydrodynamical simulations in a (256 pc)3 periodic box to model the impact of supernova (SN) explosions on the multiphase interstellar medium (ISM) for initial densities n=0.5-30cm−3 and SN rates 1-720Myr−1. We include radiative cooling, diffuse heating, and the formation of molecular gas using a chemical network. The SNe explode either at random positions, at density peaks, or both. We further present a model combining thermal energy for resolved and momentum input for unresolved SNe. Random driving at high SN rates results in hot gas (T≳106K) filling >90 per cent of the volume. This gas reaches high pressures (10450 per cent), residing in small, dense clumps. Such a model might resemble the dense ISM in high-redshift galaxies. Peak driving results in huge radiative losses, producing a filamentary ISM with virtually no hot gas, and a small molecular hydrogen mass fraction (≪1 per cent). Varying the ratio of peak to random SNe yields ISM properties in between the two extremes, with a sharp transition for equal contributions. The velocity dispersion in H i remains≲10 km s−1 in all cases. For peak driving, the velocity dispersion in Hα can be as high as 70 km s−1 due to the contribution from young, embedded SN remnant
