  We present the results from a search for gravitational-wave transients
associated with core-collapse supernovae observed optically within 30 Mpc
during the third observing run of Advanced LIGO and Advanced Virgo. No
gravitational wave associated with a core-collapse supernova has been
identified. We then report the detection efficiency for a variety of possible
gravitational-wave emissions. For neutrino-driven explosions, the distance at
which we reach 50% detection efficiency is up to 8.9 kpc, while more energetic
magnetorotationally-driven explosions are detectable at larger distances. The
distance reaches for selected models of the black hole formation, and quantum
chromodynamics phase transition are also provided. We then constrain the
core-collapse supernova engine across a wide frequency range from 50 Hz to 2
kHz. The upper limits on gravitational-wave energy and luminosity emission are
at low frequencies down to $10^{-4}\,M_\odot c^2$ and $6 \times
10^{-4}\,M_\odot c^2$/s, respectively. The upper limits on the proto-neutron
star ellipticity are down to 3 at high frequencies. Finally, by combining the
results obtained with the data from the first and second observing runs of LIGO
and Virgo, we improve the constraints of the parameter spaces of the extreme
emission models. Specifically, the proto-neutron star ellipticities for the
long-lasting bar mode model are down to 1 for long emission (1 s) at high
frequency.
