I model the multi-phase interstellar medium (ISM) randomly heated and shocked by supernovae, with gravity, differential rotation and other parameters we understand to be typical of the solar neighbourhood. The simulations are 3D extending horizontally 1 x 1 kpc squared and vertically 2 kpc, symmetric about the galactic mid-plane. They routinely span gas number densities 1/10000 to 100 per cubic cm, temperatures 100 to 100 MK, speeds up to 10000 km/s and Mach number up to 25. Radiative cooling is applied from two widely adopted parameterizations, and compared directly to assess the sensitivity of the results to cooling. There is strong evidence to describe the ISM as comprising well defined cold, warm and hot regions, which are statistically close to thermal and total pressure equilibrium. This result is not sensitive to the choice of parameters considered here. The distribution of the gas density within each can be robustly modelled as lognormal. Appropriate distinction is required between the properties of the gases in the supernova active mid-plane and the more homogeneous phases outside this region. The connection between the fractional volume of a phase and its various proxies is clarified. An exact relation is then derived between the fractional volume and the filling factors defined in terms of the volume and probabilistic averages. These results are discussed in both observational and computational contexts. The correlation scale of the random flows is calculated from the velocity autocorrelation function; it is of order 100 pc and tends to grow with distance from the mid-plane. The origin and structure of the magnetic fields in the ISM is also investigated in non-ideal MHD simulations. A seed magnetic field, with volume average of roughly 4 nG, grows exponentially to reach a statistically steady state within 1.6 Gyr. ; Comment: Doctoral Thesis 153 pages http://hdl.handle.net/10443/1755