  It is well known that the Minimal Supersymmetric Standard Model (MSSM) can
enhance Br[B_{d,s}->mu^+ mu^-] by orders of magnitude, even if we assume the
Cabibbo-Kobayashi-Maskawa (CKM) matrix to be the only source of flavour
violation. Of particular interest is the quantity R=Br[B_d->mu^+
mu^-]/Br[B_s->mu^+ mu^-], since (i) the theoretical errors cancel to a large
extent, and (ii) it offers a theoretically clean way of extracting the ratio
|V_{td}/V_{ts}| in the Standard Model, which predicts R_SM ~ |V_{td}/V_{ts}|^2
\~ O(10^{-2}). Exploring three different scenarios of modified minimal flavour
violation, we find that part of the MSSM parameter space can accommodate large
B_{d,s}->mu^+ mu^- branching fractions, while being consistent with various
experimental constraints. More importantly, we show that the ratio R can be as
large as O(1), while the individual branching fractions may be amenable to
detection by ongoing experiments. We conclude that within the MSSM with large
tan[beta] the decay rates of B_{d,s}->mu^+ mu^- be of comparable size even in
the case where flavour violation is due solely to the CKM matrix.
