  We have performed $\mu$SR studies on single crystals of SrCu$_2$(BO$_3$)$_2$,
a quasi-two-dimensional spin system with a spin singlet ground state. We
observe two different muon sites which we associate with muons located adjacent
to the two inequivalent O sites. One site, presumed to be located in the
Cu-O-Cu superexchange path, exhibits a large increase in the frequency shift
with decreasing temperature which is unaffected by the singlet formation,
indicating that the muon has locally broken the singlet bond. We have also
performed $\mu$SR on single crystals of SrMg$_{0.05}$Cu$_{1.95}$(BO$_3$)$_2$,
Sr$_{0.96}$La$_{0.04}$Cu$_2$(BO$_3$)$_2$, and
Sr$_{0.95}$Na$_{0.05}$Cu$_2$(BO$_3$)$_2$. We have found that the frequency
shifts of these doped samples are equivalent and contain three branches at low
temperatures. Two of these branches map on to the branches observed in the pure
sample reasonably well and so we attribute the third branch to effect of the
dopants. Specifically, this third branch represents the case when the muon sits
at a site in the superexchange path lacking a corresponding singlet, due to it
already being broken as a result of doping. This then leads to the conclusion
that singlets are broken in this system when it is doped both in and out of the
CuBO$_3$ planes.
