Neutrinos and antineutrinos of all three flavours are emitted during the post-bounce phase of a core-collapse supernova with having average energies more than that of They can be detected by the new earthbound detectors like SNO and Super-Kamiokande which are sensitive to neutrinos of all three flavours. In this letter we consider the effect of flavour oscillations on the neutrino flux and their expected number of events at the detector. We do a three-generation analysis and for the mass and mixing schemes we first consider the threefold maximal mixing model consistent with the solar and the atmospheric neutrino data and next a scenario with one m²~10^-11 eV² (solar range) and the other m²~10^-18 eV², for which the oscillation length is of the order of the supernova distance. In both these scenarios there are no matter effects in the resultant neutrino spectrum and one is concerned with vacuum oscillations. We find that though neutrino oscillations result in a depletion in the number of _e and coming from the supernova, the actual signals at the detectors are appreciable enhanced as the _e and energy spectra become harder.