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 at the Fréjus 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 flavour transitions with unprecedented statistics. After discussing the ingredients of our calculations, we compute neutrino event rates from inverse beta decay (), 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.