This paper presents an experimental study of different instability scenarios in a parallelogram-shaped internal wave attractor in a trapezoidal domain filled with a uniformly stratified fluid. Energy is injected into the system via the oscillatory motion of a vertical wall of the trapezoidal domain. Whole-field velocity measurements are performed with the conventional particle image velocimetry technique. In the linear regime, the total kinetic energy of the fluid system is used to quantify the strength of attractors as a function of coordinates in the parameter space defining their zone of existence, the so-called Arnold tongue. In the nonlinear regime, the choice of the operational point in the Arnold tongue is shown to have a significant impact on the scenario of the onset of triadic instability, most notably in terms of the influence of confinement on secondary waves. The onset of triadic resonance instability may occur as a spatially localized event similar to Scolan et al. (Phys. Rev. Lett., vol. 110, 2013, 234501) in the case of strong focusing or in form of growing normal modes as in McEwan (J. Fluid Mech., vol. 50, 1971, pp. 431–448) for the limiting case of rectangular domain. In the present paper, we describe also a new intermediate scenario for the case of weak focusing. We explore the long-term behaviour of cascades of triadic instabilities in wave attractors and show a persistent trend toward formation of standing-wave patterns corresponding to some discrete peaks of the frequency spectrum. At sufficiently high level of energy injection the system exhibits a ‘mixing box’ regime which has certain qualitatively universal properties regardless to the choice of the operating point in the Arnold tongue. In particular, for this regime, we observe a statistics of events with high horizontal vorticity, which serve as kinematic indicators of mixing.