  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.
