Context. General circulation models of the atmosphere of hot Jupiters have shown the existence of a supersonic eastward equatorial jet. These results have been obtained using numerical schemes that filter out vertically propagating sound waves and assume vertical hydrostatic equilibrium, or were acquired with fully compressive codes that use large dissipative coefficients. Aims: We remove these two limitations and investigate the effects of compressibility on the atmospheric dynamics by solving the standard Euler equations. Methods: This was done by means of a series of simulations performed in the framework of the equatorial β-plane approximation using the finite-volume shock-capturing code RAMSES. Results: At low resolution, we recover the classical results described in the literature: we find a strong and steady supersonic equatorial jet of a few km s-1 that displays no signature of shocks. We next show that the jet zonal velocity depends significantly on the grid meridional resolution. When this resolution is fine enough to properly resolve the jet, the latter is subject to a Kelvin-Helmholtz instability. The jet zonal mean velocity displays regular oscillations with a typical timescale of a few days and a significant amplitude of about 15% of the jet velocity. We also find compelling evidence for the development of a vertical shear instability at pressure levels of a few bars. It seems to be responsible for an increased downward kinetic energy flux that significantly affects the temperature of the deep atmosphere and appears to act as a form of drag on the equatorial jet. This instability also creates velocity fluctuations that propagate upward and steepen into weak shocks at pressure levels of a few mbars. Conclusions: We conclude that hot-Jupiter equatorial jets are potentially unstable to both a barotropic Kelvin-Helmholtz instability and a vertical shear instability. Upon confirmation using more realistic models, these two instabilities could result in significant time variability of the atmospheric winds, may provide a small-scale dissipation mechanism in the flow, and might have consequences for the internal evolution of hot Jupiters.