Despite intense scrutiny, the progenitor system(s) that gives rise to Type Ia supernovae has remained remarkably unconstrained by direct observation. The favored theory invokes a carbon-oxygen white dwarf accreting hydrogen-rich material from a close companion until a thermonuclear runaway ensues that incinerates the white dwarf. However, simulations resulting from this single-degenerate, binary channel demand the presence of low-velocity H-alpha emission in spectra taken during the late nebular phase, since a portion of the companion's envelope becomes entrained in the ejecta. This hydrogen has never been detected, but has only rarely been sought. In this talk I will present results from an ongoing campaign to obtain deep, multi-epoch, nebular-phase spectroscopy of nearby Type Ia supernovae in an effort to detect this telltale signature of the companion star. Results from two additional investigations that each seek different observational signatures resulting from popular progenitor system models will also be discussed.
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