ABSTRACT 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 3σ upper limit on the fraction of SNe Ia that are produced through the classical H-rich non-degenerate companion scenario of $\lt 5.5 {{\ \rm per\ cent}}$. Additionally, we set a tentative 3σ upper limit otan He star progenitor scenarios of $\lt 6.4 {{\ \rm per\ cent}}$, although further theoretical modelling is required. These limits refer to our most representative sample including normal, 91bg-like, 91T-like, and ‘super-Chandrasekhar’ SNe Ia 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 an SN Ia from 150 to 500 d after maximum using the peak magnitude and decline rate parameter Δm15(B).