  During the last few years the Tevatron has improved the bounds on rare
$B$-meson decays into two leptons. Sensitivity to this decay is also one of the
benchmark goals for LHCb performance. We compute the complete 1-loop MSSM
contribution to $B^0_{s,d}\to \mu^+\^-mu$ and study the predictions for
arbitrary flavour mixing parameters. We discuss the possibility of both
enhancing and suppressing the branching ratios relative to their SM
expectations. We find that there are ``cancellation regions'' in parameter
space where the branching ratio is suppressed well below the SM expectation,
making it effectively invisible to the LHC.
