  Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) is a Standard Model
process that, although predicted for decades, has only been detected recently
by the COHERENT collaboration. Now that CE$\nu$NS has been discovered, it
provides a new probe for physics beyond the Standard Model. We study the
potential to probe New Physics with CE$\nu$NS through the use of low
temperature bolometers at a reactor source. We consider contributions to
CE$\nu$NS due to a neutrino magnetic moment (NMM), Non-Standard Interactions
(NSI) that may or may not change flavor, and simplified models containing a
massive scalar or vector mediator. Targets consisting of Ge, Zn, Si, CaWO$_4$,
and Al$_2$O$_3$ are examined. We show that by reaching a percentage-level
precision measurement on the CE$\nu$NS energy spectrum down to
$\mathcal{O}(10)$ eV, forthcoming experiments will improve by two orders of
magnitude both the CE$\nu$NS-based NMM limit and the search for new massive
mediators. Additionally, we demonstrate that such dedicated low-threshold
CE$\nu$NS experiments will lead to unprecedented constraints on NSI parameters
(particularly when multiple targets are combined) which will have major
implications for the global neutrino physics program.
