The determination of the brown dwarf binary fraction may contribute to the understanding of the substellar formation mechanisms. Unresolved brown dwarf binaries may be revealed through their peculiar spectra or the discrepancy between optical and near-infrared spectral-type classification. We obtained medium-resolution spectra of 22 brown dwarfs with these characteristics using the X-Shooter spectrograph at the Very Large Telescope. We aimed to identify brown dwarf binary candidates, and to test if the BT-Settl 2014 atmospheric models reproduce their observed spectra. To find binaries spanning the L-T boundary, we used spectral indices and compared the spectra of the selected candidates to single spectra and synthetic binary spectra. We used synthetic binary spectra with components of same spectral type to determine as well the sensitivity of the method to this class of binaries. We identified three candidates to be combination of L plus T brown dwarfs. We are not able to identify binaries with components of similar spectral type. In our sample, we measured minimum binary fraction of 9.1^{+9.9}_{-3.0} per cent. From the best fit of the BT-Settl models 2014 to the observed spectra, we derived the atmospheric parameters for the single objects. The BT-Settl models were able to reproduce the majority of the spectral energy distributions from our objects, and the variation of the equivalent width of the Rb I (794.8 nm) and Cs I (852.0 nm) lines with the spectral type. None the less, these models did not reproduce the evolution of the equivalent widths of the Na I (818.3 and 819.5 nm) and K I (1253 nm) lines with the spectral type.