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The Muon Scattering Experiment at the Paul Scherrer Institut uses a mixed beam of electrons, muons, and pions, necessitating precise timing to identify the beam particles and reactions they cause. We describe the design and performance of three timing detectors using plastic scintillator read out with silicon photomultipliers that have been built for the experiment. The Beam Hodoscope, upstream of the scattering target, counts the beam flux and precisely times beam particles both to identify species and provide a starting time for time-of-flight measurements. The Beam Monitor, downstream of the scattering target, counts the unscattered beam flux, helps identify background in scattering events, and precisely times beam particles for time-of-flight measurements. The Beam Focus Monitor, mounted on the target ladder under the liquid hydrogen target inside the target vacuum chamber, is used in dedicated runs to sample the beam spot at three points near the target center, where the beam should be focused.