  We present deep Hubble Space Telescope imaging at the locations of four,
potentially hostless, long-faded Type Ia supernovae (SNe Ia) in low-redshift,
rich galaxy clusters that were identified in the Multi-Epoch Nearby Cluster
Survey. Assuming a steep faint-end slope for the galaxy cluster luminosity
function ($\alpha_d=-1.5$), our data includes all but $\lesssim0.2\%$ percent
of the stellar mass in cluster galaxies ($\lesssim0.005\%$ with
$\alpha_d=-1.0$), a factor of 10 better than our ground-based imaging. Two of
the four SNe Ia still have no possible host galaxy associated with them
($M_R>-9.2$), confirming that their progenitors belong to the intracluster
stellar population. The third SNe Ia appears near a faint disk galaxy
($M_V=-12.2$) which has a relatively high probability of being a chance
alignment. A faint, red, point source coincident with the fourth SN Ia's
explosion position ($M_V=-8.4$) may be either a globular cluster (GC) or faint
dwarf galaxy. We estimate the local surface densities of GCs and dwarfs to show
that a GC is more likely, due to the proximity of an elliptical galaxy, but
neither can be ruled out. This faint host implies that the SN Ia rate in dwarfs
or GCs may be enhanced, but remains within previous observational constraints.
We demonstrate that our results do not preclude the use of SNe Ia as bright
tracers of intracluster light at higher redshifts, but that it will be
necessary to first refine the constraints on their rate in dwarfs and GCs with
deep imaging for a larger sample of low-redshift, apparently hostless SNe Ia.
