We present a MUSE (Multi-Unit Spectroscopic Explorer) and KMOS (K-band Multi-Object Spectrograph) dynamical study 405 star-forming galaxies at redshift z = 0.28–1.65 (median redshift z¯ = 0.84). Our sample is representative of the star-forming ‘main sequence’, with star formation rates of SFR = 0.1–30 M yr−1 and stellar masses M = 108–1011 M. For 49 ± 4 per cent of our sample, the dynamics suggest rotational support, 24 ± 3 per cent are unresolved systems and 5 ± 2 per cent appear to be early-stage major mergers with components on 8–30 kpc scales. The remaining 22 ± 5 per cent appear to be dynamically complex, irregular (or face-on systems). For galaxies whose dynamics suggest rotational support, we derive inclination-corrected rotational velocities and show that these systems lie on a similar scaling between stellar mass and specific angular momentum as local spirals with j = J/M ∝ M2/3 but with a redshift evolution that scales as j ∝ M2/3 (1 + z) −1. We also identify a correlation between specific angular momentum and disc stability such that galaxies with the highest specific angular momentum (log(j/M2/3 ) > 2.5) are the most stable, with Toomre Q = 1.10 ± 0.18, compared to Q = 0.53 ± 0.22 for galaxies with log(j/M2/3 ) < 2.5. At a fixed mass, the Hubble Space Telescope morphologies of galaxies with the highest specific angular momentum resemble spiral galaxies, whilst those with low specific angular momentum are morphologically complex and dominated by several bright star-forming regions. This suggests that angular momentum plays a major role in defining the stability of gas discs: atz∼1, massive galaxies that have discs with low specific angular momentum are globally unstable, clumpy and turbulent systems. In contrast, galaxies with high specific angular momentum have evolved into stable discs with spiral structure where star formation is a local (rather than global) process.