Rings are distinctive features of many disc galaxies and their location and properties are closely related to the disc dynamics. In particular, rings are often associated to stellar bars, but the details of this connection are far from clear. We have studied the frequency and dimensions of inner and outer rings in the local Universe as a function of disc parameters and the amplitude of non-axisymmetries. We used the 1320 not highly inclined disc galaxies (i < 65°) from the S4G survey. The ring fraction increases with bar Fourier density amplitude: this can be interpreted as evidence for the role of bars in ring formation. The sizes of inner rings are positively correlated with bar strength: this can be linked to the radial displacement of the 1/4 ultraharmonic resonance while the bar grows and the pattern speed decreases. The ring's intrinsic ellipticity is weakly controlled by the non-axisymmetric perturbation strength: this relation is not as strong as expected from simulations, especially when we include the dark matter halo in the force calculation. The ratio of outer-to-inner ring semi-major axes is uncorrelated with bar strength: this questions the manifold origin of rings. In addition, we confirm that (i) ˜1/3 (˜1/4) of the galaxies hosting inner (outer) rings are not barred; (ii) on average, the sizes and shapes of rings are roughly the same for barred and non-barred galaxies; and (iii) the fraction of inner (outer) rings is a factor of 1.2-1.4 (1.65-1.9) larger in barred galaxies than in their non-barred counterparts. Finally, we apply unsupervised machine learning (self-organising maps, SOMs) to show that, among early-type galaxies, ringed or barred galaxies cannot be univocally distinguished based on 20 internal and external fundamental parameters. We confirm, with the aid of SOMs, that rings are mainly hosted by red, massive, gas-deficient, dark-matter poor, and centrally concentrated galaxies. We conclude that the present-day coupling between rings and bars is not as robust as predicted by numerical models, and diverse physical mechanisms and timescales determine ring formation and evolution.