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ABSTRACT The observed diversity in Type Ia supernovae (SNe Ia) – the thermonuclear explosions of carbon–oxygen white dwarf stars used as cosmological standard candles – is currently met with a variety of explosion models and progenitor scenarios. To help improve our understanding of whether and how often different models contribute to the occurrence of SNe Ia and their assorted properties, we present a comprehensive analysis of seven nearby SNe Ia. We obtained one to two epochs of optical spectra with Gemini Observatory during the nebular phase (>200 d past peak) for each of these events, all of which had time series of photometry and spectroscopy at early times (the first ∼8 weeks after explosion). We use the combination of early- and late-time observations to assess the predictions of various models for the explosion (e.g. double-detonation, off-centre detonation, stellar collisions), progenitor star (e.g. ejecta mass, metallicity), and binary companion (e.g. another white dwarf or a non-degenerate star). Overall, we find general consistency in our observations with spherically symmetric models for SN Ia explosions, and with scenarios in which the binary companion is another degenerate star. We also present an in-depth analysis of SN 2017fzw, a member of the subgroup of SNe Ia which appear to be transitional between the subluminous ‘91bg-like’ events and normal SNe Ia, and for which nebular-phase spectra are rare.