Context. The presence of sub-micron grains has been inferred in several debris discs, usually because of a blue colour of the spectrum in scattered light or a pronounced silicate band around 10 μm, even though these particles should be blown out by stellar radiation pressure on very short timescales. So far, no fully satisfying explanation has been found for this apparent paradox. Aims. We investigate the possibility that the observed abundances of sub-micron grains could be naturally produced in bright debris discs, where the high collisional activity produces them at a rate high enough to partially compensate for their rapid removal. We also investigate to what extent this potential presence of small grains can affect our understanding of some debris disc characteristics. Methods. We used a numerical collisional code to follow the collisional evolution of a debris disc down to sub-micron grains far below the limiting blow-out size s_blow. We considered compact astrosilicates and explored different configurations: A and G stars, cold and warm discs, bright and very bright systems. We then produced synthetic spectra and spectral energy distributions, where we identified and quantified the signature of unbound sub-micron grains. Results. We find that in bright discs (fractional luminosity ≳10⁻³) around A stars, the number of sub-micron grains is always high enough to leave detectable signatures in scattered light where the disc colour becomes blue, and also in the mid-IR (10 ≲ λ ≲ 20 μm), where they boost the disc luminosity by at least a factor of 2 and induce a pronounced silicate solid-state band around 10 μm. We also show that with this additional contribution of sub-micron grains, the spectral energy distribution can mimic that of two debris belts separated by a factor of ~2 in radial distance. For G stars, the effect of s ≤ s_blow grains remains limited in the spectra although they dominate the geometrical cross section of the system. We also find that for all considered cases, the halo of small (bound and unbound) grains that extends far beyond the main disc contributes to ~50% of the flux up to λ ~ 50 μm wavelengths.