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.
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