  Magnetic fields grow quickly even at early cosmological times, suggesting the
action of a small-scale dynamo (SSD) in the interstellar medium of galaxies.
Many studies have focused on idealized turbulent driving of the SSD. Here we
simulate more realistic supernova-driven turbulence to determine whether it can
drive an SSD. Magnetic field growth occurring in our models appears
inconsistent with simple tangling of magnetic fields, but consistent with SSD
action, reproducing and confirming models by Balsara et al. (2004) that did not
include physical resistivity $\eta$. We vary $\eta$, as well as the numerical
resolution and supernova rate, $\dot\sigma$, to delineate the regime in which
an SSD occurs. For a given $\dot\sigma$ we find convergence for SSD growth rate
with resolution of a parsec. For $\dot\sigma\simeq\dot\sigma_{\rm sn}$, with
$\dot\sigma_{\rm sn}$ the solar neighbourhood rate, the critical resistivity
below which an SSD occurs is $0.005>\eta_{\rm crit}>0.001\,\rm kpc^{-1}\,\rm km
s^{-1}$, and this increases with the supernova rate. Across the modelled range
of 0.5--4 pc resolution we find that for $\eta<\eta_{\rm crit}$, the SSD
saturates at about 5% of kinetic energy equipartition, independent of growth
rate. In the range $0.2\dot\sigma_{\rm sn}\leq \dot\sigma\leq8\dot\sigma_{\rm
sn}$ growth rate increases with $\dot\sigma$. SSDs in the supernova-driven
interstellar medium commonly exhibit erratic growth.
