  We place statistical constraints on Type Ia supernova (SN Ia) progenitors
using 227 nebular phase spectra of 111 SNe Ia. We find no evidence of stripped
companion emission in any of the nebular phase spectra. Upper limits are placed
on the amount of mass that could go undetected in each spectrum using recent
hydrodynamic simulations. With these null detections, we place an observational
$3\sigma$ upper limit on the fraction of SNe Ia that are produced through the
classical H-rich non-degenerate companion scenario of < 5.5%. Additionally, we
set a tentative $3\sigma$ upper limit on He star progenitor scenarios of <
6.4%, although further theoretical modelling is required. These limits refer to
our most representative sample including normal, 91bg-like, 91T-like, and
"Super Chandrasekhar" \sne but excluding SNe Iax and SNe Ia-CSM. As part of our
analysis, we also derive a Nebular Phase Phillips Relation, which approximates
the brightness of a SN Ia from $150-500$~days after maximum using the peak
magnitude and decline rate parameter $\Delta m_{15} (B)$.
