The intermittent small-scale structure of turbulence governs energy dissipation in many astrophysical plasmas and is often believed to have universal properties for sufficiently large systems. In this work, we argue that small-scale turbulence in accretion discs is universal in the sense that it is insensitive to the magnetorotational instability (MRI) and background shear, and therefore indistinguishable from standard homogeneous magnetohydrodynamic (MHD) turbulence at small scales. We investigate the intermittency of current density, vorticity and energy dissipation in numerical simulations of incompressible MHD turbulence driven by the MRI in a shearing box. We find that the simulations exhibit a similar degree of intermittency as in standard MHD turbulence. We perform a statistical analysis of intermittent dissipative structures and find that energy dissipation is concentrated in thin sheet-like structures that span a wide range of scales up to the box size. We show that these structures exhibit strikingly similar statistical properties to those in standard MHD turbulence. Additionally, the structures are oriented in the toroidal direction with a characteristic tilt of approximately 17$$_{.}^{\circ}$$5, implying an effective guide field in that direction.