  We reconsider the impact of $\bar\nu_e \leftrightarrow \bar\nu_{\mu,\tau}$
neutrino oscillations on the observed $\bar\nu_e$ signal of supernova SN 1987A.
Performing a maximum-likelihood analysis using as fit parameters the released
binding energy $\Eb$ and the average neutrino energy $\Ee$, we find as previous
analyses that $\bar\nu_e \leftrightarrow \bar\nu_{\mu,\tau}$ oscillations with
large mixing angles have lower best-fit values for $\Ee$ than small-mixing
angle (SMA) oscillations. Moreover, the inferred value of $\Ee$ is already in
the SMA case lower than those found in simulations. This apparent conflict has
been interpreted as evidence against the large mixing oscillation solutions to
the solar neutrino problem. In order to quantify the degree to which the
experimental data favour the SMA over the large mixing solutions we use their
likelihood ratios as well as a Kolmogorov-Smirnov test. We find within the
range of SN parameters predicted by simulations regions in which the LMA-MSW
solution is either only marginally disfavoured or favoured compared to the
SMA-MSW solution. We conclude therefore that the LMA-MSW solution is not in
conflict with the current understanding of SN physics. In contrast, the vacuum
oscillation and the LOW solutions to the solar neutrino problem can be excluded
at the $4\sigma$ level for most of the SN parameter ranges found in
simulations. Only a marginal region with low values of $\Ee$,
$<E_{\bar\nu_{\mu,\tau}}>$ and $\Eb$ is left over, in which these oscillation
solutions can be reconciled with the neutrino signal of SN 1987A.
