We discuss big bang nucleosynthesis constraints on maximal ${\nu }_{\mu }\leftrightarrow {\nu }_s$ mixing. Vacuum ${\nu }_{\mu }\leftrightarrow {\nu }_s$ oscillation has been proposed as one possible explanation of the Super Kamiokande atmospheric neutrino data. Based on the most recent primordial abundance measurements, we find that the effective number of neutrino species for big bang nucleosynthesis (BBN) is $N_{\nu }\lesssim \mathrm{3.3.}$ Assuming that all three active neutrinos are light (with masses $\ll 1\mathrm{MeV}$), we examine BBN constraints on ${\nu }_{\mu }\leftrightarrow {\nu }_s$ mixing in two scenarios: (1) a negligible lepton asymmetry (the standard picture) and (2) the presence of a large lepton asymmetry which has resulted from an amplification by ${\nu }_{\tau }\leftrightarrow {\nu }_{s^{\prime }}$ mixing $({\nu }_{s^{\prime }}$ being ${\nu }_s$ or another sterile neutrino species). The latter scenario has been proposed recently to reconcile the BBN constraints and large-angle ${\nu }_{\mu }\leftrightarrow {\nu }_s$ mixing. We find that the large-angle ${\nu }_{\mu }\leftrightarrow {\nu }_s$ mixing in the first scenario, which would yield $N_{\nu }\approx 4,$ is ruled out as an explanation of the Super Kamiokande data. It is conceivably possible for the ${\nu }_{\mu }\leftrightarrow {\nu }_s$ solution to evade BBN bounds in the second scenario, but only if $200{\mathrm{eV}}^2\lesssim m_{{\nu }_{\tau }}^2-m_{{\nu }_{s^{\prime }}}^2\lesssim {10}^4{\mathrm{eV}}^2$ is satisfied, and if ${\nu }_{\tau }$ decays non-radiatively with a lifetime $\lesssim {10}^3\mathrm{yr}.$ This mass-squared difference implies $15\mathrm{eV}\lesssim m_{{\nu }_{\tau }}\lesssim 100\mathrm{eV}$ if ${\nu }_{s^{\prime }}$ is much lighter than ${\nu }_{\tau }.$ We conclude that maximal (or near maximal) ${\nu }_{\mu }\leftrightarrow {\nu }_{\tau }$ mixing is a more likely explanation of the Super Kamiokande data.

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