Magnetohydrodynamic (MHD) phenomena occurring in the ultra-low safety factor (ULq) configuration are investigated by means of 3D nonlinear MHD simulations. The ULq configuration, a screw pinch characterized by the edge safety factor qedge in the interval 0 < qedge < 1, is the intermediate state between the tokamak and the reversed field pinch. This numerical study, based on the simple frame of the visco-resistive pressureless MHD model, shows that ULq plasmas have the natural tendency to select discrete qedge values which are about the major rational numbers, suggesting plasma self-organization. Similar behaviour is observed in experimental ULq discharges, like those recently obtained exploiting the flexibility of the RFX-mod device. The transition of qedge from a major rational number to the next one occurs together with the development of a kink deformation of the plasma column, whose stabilization yields a nearly axisymmetric state with a rather flat q profile. Numerical simulations also show that it is possible to sustain either of the two conditions, namely, the saturated kink helical configuration and the axisymmetric one, by forcing qedge at a suitable value. Finally, the effects of this MHD phenomenology on the confinement properties of ULq plasmas are discussed.