Supernovae are among the most powerful and influential explosions in the universe. They are also ideal multi-messenger laboratories to study extreme astrophysics. However, many fundamental properties of supernovae related to their diverse progenitor systems and explosion mechanisms remain poorly constrained. Here we outline how late-time spectroscopic observations obtained during the nebular phase (several months to years after explosion), made possible with the next generation of Extremely Large Telescopes, will facilitate transformational science opportunities and rapidly accelerate the community towards our goal of achieving a complete understanding of supernova explosions. We highlight specific examples of how complementary GMT and TMT instrumentation will enable high fidelity spectroscopy from which the line profiles and luminosities of elements tracing mass loss and ejecta can be used to extract kinematic and chemical information with unprecedented detail, for hundreds of objects. This will provide uniquely powerful constraints on the evolutionary phases stars may experience approaching a supernova explosion; the subsequent explosion dynamics; their nucleosynthesis yields; and the formation of compact objects that may act as central engines.
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