  Neutrinos produced in the hot and dense interior of the next galactic
supernova would be visible at dark matter experiments in coherent elastic
nuclear recoils. While studies on this channel have focused on successful
core-collapse supernovae, a thermonuclear (Type Ia) explosion, or a
core-collapse that fails to explode and forms a black hole, are as likely to
occur as the next galactic supernova event. I show that generation-3 noble
liquid-based dark matter experiments such as DARWIN and ARGO, operating at
sub-keV thresholds with ionization-only signals, would distinguish between (a)
leading hypotheses of Type Ia explosion mechanisms by detecting an
$\mathcal{O}$(1) s burst of $\mathcal{O}$(1) MeV neutrinos, and (b) progenitor
models of failed supernovae by detecting an $\mathcal{O}$(1) s burst of
$\mathcal{O}$(10) MeV neutrinos, especially by marking the instant of black
hole formation from abrupt stoppage of neutrino detection. This detection is
sensitive to all neutrino flavors and insensitive to neutrino oscillations,
thereby making measurements complementary to neutrino experiments.
