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We present the results of a radial velocity (RV) survey of 14 brown dwarfs (BDs) and very low-mass (VLM) stars in the Upper Scorpius OB association (UScoOB) and 3 BD candidates in the rho Ophiuchi dark cloud core. We obtained high-resolution echelle spectra at the Very Large Telescope using Ultraviolet and Visual Echelle Spectrograph (UVES) at two different epochs for each object, and measured the shifts in their RVs to identify candidates for binary/multiple systems in the sample. The average time separation of the RV measurements is 21.6d, and our survey is sensitive to the binaries with separation < 0.1 au. We found that 4 out of 17 objects (or 24^{+16}_{-13} per cent by fraction) show a significant RV change in 4-33d time scale, and are considered as binary/multiple `candidates.' We found no double-lined spectroscopic binaries in our sample, based on the shape of cross-correlation curves. The RV dispersion of the objects in UScoOB is found to be very similar to that of the BD and VLM stars in Chamaeleon I (Cha I). We also found the distribution of the mean rotational velocities (v sin i) of the UScoOB objects is similar to that of the Cha I, but the dispersion of v sin i is much larger than that of the Cha I objects.