  Several current projects aim at building a large water-Cherenkov detector,
with a fiducial volume about 20 times larger than in the current
Super-Kamiokande experiment. These projects include the Underground nucleon
decay and Neutrino Observatory (UNO) in the Henderson Mine (Colorado), the
Hyper-Kamiokande (HK) detector in the Tochibora Mine (Japan), and the MEgaton
class PHYSics (MEMPHYS) detector in the Frejus site (Europe). We study the
physics potential of a reference next-generation detector (0.4 Mton of fiducial
mass) in providing information on supernova neutrino flavor transitions with
unprecedented statistics. After discussing the ingredients of our calculations,
we compute neutrino event rates from inverse beta decay ($\bar\nu_e p\to e^+ n
$), elastic scattering on electrons, and scattering on oxygen, with emphasis on
their time spectra, which may encode combined information on neutrino
oscillation parameters and on supernova forward (and possibly reverse) shock
waves. In particular, we show that an appropriate ratio of low-to-high energy
events can faithfully monitor the time evolution of the neutrino crossing
probability along the shock-wave profile. We also discuss some background
issues related to the detection of supernova relic neutrinos, with and without
the addition of gadolinium.
