  We report a muon spin rotation/relaxation ($\mu$SR) study of
single-crystalline samples of the $\alpha$-RuCl$_3$ honeycomb magnet, which is
presumed to be a model compound for the Kitaev-Heisenberg interaction. It is
inferred from magnetic susceptibility and specific-heat measurements that the
present samples exhibit successive magnetic transitions at different critical
temperatures $T_{\rm N}$ with decreasing temperature, eventually falling into
the $T_{\rm N}=7$ K antiferromagnetic (7 K) phase that has been observed in
only single-crystalline specimens with the least stacking fault. Via $\mu$SR
measurements conducted under a zero external field, we show that such behavior
originates from a phase separation induced by the honeycomb plane stacking
fault, yielding multiple domains with different $T_{\rm N}$'s. We also perform
$\mu$SR measurements under a transverse field in the paramagnetic phase to
identify the muon site from the muon-Ru hyperfine parameters. Based on a
comparison of the experimental and calculated internal fields at the muon site
for the two possible spin structures inferred from neutron diffraction data, we
suggest a modulated zig-zag spin structure for the 7 K phase, with the
amplitude of the ordered magnetic moment being significantly reduced from that
expected for the orbital quenched spin-1/2 state.
