  We study the impact of neutrino oscillations on the interpretation of the
supernova (SN) 1987A neutrino signal by means of a maximum-likelihood analysis.
We focus on oscillations between $\overline\nu_e$ with $\overline\nu_\mu$ or
$\overline\nu_\tau$ with those mixing parameters that would solve the solar
neutrino problem. For the small-angle MSW solution ($\Delta
m^2\approx10^{-5}\,\rm eV^2$, $\sin^22\Theta_0\approx0.007$), there are no
significant oscillation effects on the Kelvin-Helmholtz cooling signal; we
confirm previous best-fit values for the neutron-star binding energy and
average spectral $\overline\nu_e$ temperature. There is only marginal overlap
between the upper end of the 95.4\% CL inferred range of $\langle
E_{\overline\nu_e}\rangle$ and the lower end of the range of theoretical
predictions. Any admixture of the stiffer $\overline\nu_\mu$ spectrum by
oscillations aggravates the conflict between experimentally inferred and
theoretically predicted spectral properties. For mixing parameters in the
neighborhood of the large-angle MSW solution ($\Delta m^2\approx10^{-5}\,\rm
eV^2$, $\sin^22\Theta_0\approx0.7$) the oscillations in the SN are adiabatic,
but one needs to include the regeneration effect in the Earth which causes the
Kamiokande and IMB detectors to observe different $\overline\nu_e$ spectra. For
the solar vacuum solution ($\Delta m^2\approx10^{-10}\,\rm eV^2$,
$\sin^22\Theta_0\approx1$) the oscillations in the SN are nonadiabatic; vacuum
oscillations take place between the SN and the detector. If either of the
large-angle solutions were borne out by the upcoming round of solar neutrino
experiments, one would have to conclude that the SN~1987A $\overline\nu_\mu$
and/or $\overline\nu_e$ spectra had been much softer than predicted by current
