ABSTRACT Vertically thickened bars, observed in the form of boxy/peanut (B/P) bulges, are found in the majority of massive barred disc galaxies in the local Universe, including our own. B/P bulges indicate that their host bars have suffered violent bending instabilities driven by anisotropic velocity distributions. We investigate for the first time how the frequency of B/P bulges in barred galaxies evolves from z = 1 to z ≈ 0, using a large sample of non-edge-on galaxies with masses M⋆ > 1010 M⊙, selected from the HST COSMOS survey. We find the observed fraction increases from $0^{+3.6}_{-0.0}{{\ \rm per\ cent}}$ at z = 1 to $37.8^{+5.4}_{-5.1}$ per cent at z = 0.2. We account for problems identifying B/P bulges in galaxies with low inclinations and unfavourable bar orientations, and due to redshift-dependent observational biases with the help of a sample from the Sloan Digital Sky Survey, matched in resolution, rest-frame band, signal-to-noise ratio and stellar mass and analysed in the same fashion. From this, we estimate that the true fraction of barred galaxies with B/P bulges increases from ∼10 per cent at z ≈ 1 to $∼ 70{{\ \rm per\ cent}}$ at z = 0. In agreement with previous results for nearby galaxies, we find a strong dependence of the presence of a B/P bulge on galaxy stellar mass. This trend is observed in both local and high-redshift galaxies, indicating that it is an important indicator of vertical instabilities across a large fraction of the age of the Universe. We propose that galaxy formation processes regulate the thickness of galaxy discs, which in turn affect which galaxies experience violent bending instabilities of the bar.