  Active galactic nuclei (AGN) are prominent environments for stellar capture,
growth and formation. These environments may catalyze stellar mergers and
explosive transients, such as thermonuclear and core-collapse supernovae (SNe).
SN explosions in AGN discs generate strong shocks, leading to unique observable
signatures. We develop an analytical model which follows the evolution of the
shock propagating in the disc until it eventually breaks out. We derive the
peak luminosity, bolometric lightcurve, and breakout time. The peak
luminosities may exceed $10^{45}$ erg s$^{-1}$ and last from hours to days. The
brightest explosions occur in regions of reduced density; either off-plane, or
in discs around low-mass central black holes ($\sim 10^6\ M_\odot$), or in
starved subluminous AGNs. Explosions in the latter two sites are easier to
observe due to a reduced AGN background luminosity. We perform suites of 1D
Lagrangian radiative hydrodynamics SNEC code simulations to validate our
results and obtain the luminosity in different bands, and 2D axisymmetric
Eulerian hydrodynamics code HORMONE simulations to study the morphology of the
ejecta and its deviation from spherical symmetry. The observed signature is
expected to be a bright blue, UV, or X-ray flare on top of the AGN luminosity
from the initial shock breakout, while the subsequent red part of the
lightcurve will largely be unobservable. We estimate the upper limit for the
total event rate to be $\mathcal{R}\lesssim 100\ \rm yr^{-1}\ Gpc^{-3}$ for
optimal conditions and discuss the large uncertainties in this estimate. Future
high-cadence transient searches may reveal these events. Some existing tidal
disruption event candidates may originate from AGN supernovae.
