We investigate the generation of vorticity in supernova driven interstellar turbulence using a local three-dimensional MHD model. Our model includes the effects of density stratification, compressibility, magnetic fields, large-scale shear due to galactic differential rotation, heating via supernova explosions and parameterized radiative cooling of the interstellar medium; we also include viscosity and resistivity. We allow for multiple supernovae, which are distributed randomly in the galactic disc and exponentially in the vertical direction. When supernovae are infrequent, so that there is no interactions between supernova remnants, the dynamics of the system is dominated by strong shocks driven by the young remnants. Supernova interactions, where shock fronts from younger remnants encounter the dense shells of the older remnants, were found to produce vorticity via the baroclinic effect. Vorticity generated by the baroclinic effect was observed to be amplified by the stretching of vortex lines, these two vorticity production mechanisms being of equal importance after 1.5 × 108 years. Motions driven by the supernova explosions also amplify the magnetic field via stretching and compression. This generates a random component from a uniform azimuthal magnetic field prescribed as an initial condition and maintains it against Ohmic losses.