  The MiniBooNE experiment at Fermilab has amassed the largest sample to date
of $\pi^0$s produced in neutral current (NC) neutrino-nucleus interactions at
low energy. This paper reports a measurement of the momentum distribution of
$\pi^0$s produced in mineral oil (CH$_2$) and the first observation of coherent
$\pi^0$ production below 2 GeV. In the forward direction, the yield of events
observed above the expectation for resonant production is attributed primarily
to coherent production off carbon, but may also include a small contribution
from diffractive production on hydrogen. Integrated over the MiniBooNE neutrino
flux, the sum of the NC coherent and diffractive modes is found to be (19.5
$\pm$1.1 (stat) $\pm$2.5 (sys))% of all exclusive NC $\pi^0$ production at
MiniBooNE. These measurements are of immediate utility because they quantify an
important background to MiniBooNE's search for $\nu_{\mu} \to \nu_e$
oscillations.
