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In this dissertation, we consider the origin of thermonuclear supernovae, known by their observational classification as type Ia (hereafter SNe Ia). In particular, we develop an entirely new means to test the ``single-degenerate'' hypothesis, in which the progenitors of these tremendous explosions are suggested to be hot and luminous accreting white dwarfs. We then strongly constrain the role of any such ``hot-mode'' SN Ia progenitor channel using both a population-based argument and an individual case study, before concluding with some more general considerations of nebulae ionized by accreting white dwarfs. \n\nType Ia supernovae have now been the subject of intensive study for decades, particularly in light of their role as standard(-izable) candles in measuring cosmological distances. However, there remains no consensus model for the evolutionary channel(s) by which they originate. In the so-called ``double-degenerate'' scenario, a binary pair of white dwarfs shed angular momentum through gravitational-wave radiation, until they inspiral and merge, triggering an explosion. Alternatively, in the classic picture of the single-degenerate scenario, a white dwarf accretes hydrogen-rich material from some main sequence or red giant companion, and grows through nuclear burning of this material at its surface until reaching sufficient mass to trigger an explosion. This suggests that single-degenerate progenitors should be extremely luminous sources in the EUV and soft X-ray bands during the accretion phase (lasting $\\sim 10^{5}$--$10^{6}$ years). For this reason, such objects are generally associated with observed ``supersoft X-ray sources'' (SSSs). Previous efforts to detect or constrain the role of any such channel have focused on detecting these objects directly in the soft X-ray band (photon energies in the range 0.3 -- 0.7 keV), either on an individual basis or as the combined emission of a diffuse population. Such an approach has yielded important constraints, but only if white dwarfs accrete principally at very high temperatures (T $\\sim$ $5\\times 10^{5}$K). However, observed SSSs are understood to lie in a broad range of temperatures, with a possible range of at least $2 \\times 10^{5}$--$10^{6}$K, and some theoretical models suggest even lower temperatures are possible. This necessitates the development of an alternative, complimentary test which can constrain the luminosity of accreting white dwarfs across a wider range of photospheric temperatures.\n\nIn this work, we demonstrate that if the single-degenerate model is correct, then accreting, nuclear-burning white dwarfs should provide the dominant source of ionizing radiation in passively-evolving galaxies, roughly 40\\% of which are known to host extended low-ionization emission-line regions (so-called ``retired'' galaxies, i.e. emission-line galaxies without either a central AGN or significant ongoing star formation). Therefore, one can search for the presence of any high-temperature single-degenerate progenitor population in these galaxies by looking for emission lines characteristic of ionization by very high-temperature ($10^{5}$ K -- $10^{6}$ K) sources. In particular, we find that recombination lines of He II, and forbidden lines of [N I] and [O I], provide the most sensitive diagnostics in retired galaxies to assess the role of accreting white dwarfs as SN Ia progenitors in any ``{\\bf hot}-mode'' (T $\\gtrsim 1.2\\times 10^{5}$K) accretion regime.\n\nFollowing this, we limit the contribution of any high-temperature single-degenerate channel to the SN Ia rate at relatively early delay-times (1 Gyr $\\leq$ t $\\leq$ 4 Gyr) to $<$~5--10\\% (for T$\\gtrsim 1.2\\times 10^{5}$K) using He II 4686\\AA\\ and [O I] 6300\\AA\\ measurements from a stacked sample (provided by Dr Jonas Johansson) of several thousand retired galaxies in the Sloan Digital Sky Survey. We also discuss how these constraints, as well as the observed soft X-ray emission of several nearby galaxies, reveal fundamental problems in our present understanding of the population synthesis of SSSs and other accreting white dwarf binaries.\n\nWe then revise the standard picture for the observational appearance of nebulae ionized by individual accreting white dwarfs, accounting for a more realistic assessment of the typical ISM densities in which such objects are likely embedded. We then provide the first formal justification for why so few SSS nebulae have been detected thus far, and demonstrate that a complete survey is within the means of modern large-aperture telescopes (such as ESO's Very Large Telescope). We then show how this approach can be extended to individual SNe Ia, by searching for fossil nebulae in the vicinity of nearby events. In particular, we use an archival pre-explosion narrow-band H$\\alpha$ + [N II] image of the vicinity of SN2014J to place constraints on the luminosity of any putative high-temperature progenitor for SN2014J (such as an accreting white dwarf).