We present the results of photometric and spectroscopic follow-ups of the lowest mass member candidates in the nearest OB association, Upper Scorpius (∼5-10 Myr; 145 ± 17 pc), with the Gran Telescopio de Canarias (GTC) and European Southern Observatory (ESO) Very Large Telescope (VLT). We confirm the membership of the large majority (>80 per cent) of candidates originally selected photometrically and astrometrically based on their spectroscopic features, weak equivalent widths of gravity-sensitive doublets and radial velocities. Confirmed members follow a sequence over a wide magnitude range (J = 17.0-19.3 mag) in several colour-magnitude diagrams with optical, near- and mid-infrared photometry and have near-infrared spectral types in the L1-L7 interval with likely masses below 15 Jupiter masses. We find that optical spectral types tend to be earlier than near-infrared spectral types by a few subclasses for spectral types later than M9. We investigate the behaviour of spectral indices, defined in the literature as a function of spectral type and gravity, by comparison with values reported in the literature for young and old dwarfs. We also derive effective temperatures in the 1900-1600 K range from fits of synthetic model-atmosphere spectra to the observed photometry, but we caution that the procedure carries large uncertainties. We determine bolometric corrections for young L dwarfs with ages of ∼5-10 Myr (Upper Sco association) and find them to be similar in the J band but larger by 0.1-0.4 mag in the K band with respect to field L dwarfs. Finally, we discover two faint young L dwarfs, Visible and Infrared Survey Telescope for Astronomy (VISTA) J1607-2146 (L4.5) and VISTA J1611-2215 (L5), that have Hα emission and possible flux excesses at 4.5 μm, pointing to the presence of accretion from a disc on to the central objects of mass below ∼15MJup at an age of 5-10 Myr.