The discovery of integrability in planar N=4 SYM theory led to considerable advances in the computation of its planar anomalous dimension spectrum. Less is known at the non-planar level where the theory is assumed to be non-integrable. I will show how statistical properties of numerical anomalous dimension spectra can give insight into the symmetries of the underlying model and that the N=4 SYM non-planar spectrum and its beta-deformed version are well described by random matrix theory, indicating their quantum-chaotic nature. Doing so I will also discuss on-going work on using on-shell methods to obtain the dilatation operator for deformed versions of N=4 SYM theory. [for zoom link please contact jung-wook(dot)kim(at)qmul(dot)ac(dot)uk]