We introduce two new concepts, frictional rigidity percolation and minimal rigidity proliferation, to help identify the nature of the frictional jamming transition as well as significantly broaden the scope of rigidity percolation. For frictional rigidity percolation, we construct rigid clusters in two different lattice models using a $(3,3)$ pebble game, while taking into account contacts below and at the Coulomb threshold. The first lattice is a honeycomb lattice with next-nearest neighbors, the second, a hierarchical lattice. For both, we generally find a continuous rigidity transition. Our numerical results suggest that, for the honeycomb lattice, the exponents associated with the transition found with the frictional $(3,3)$ pebble game are distinct from those of a central-force $(2,3)$ pebble game. We propose that localized motifs, such as hinges, connecting rigid clusters that are allowed only with friction could give rise to this new frictional universality class. However, the closeness of the order parameter exponent between the two cases hints at potential superuniversality. To explore this possibility, we construct a bespoke cluster generating algorithm invoking generalized Henneberg moves, dubbed minimal rigidity proliferation. The minimally rigid clusters the algorithm generates appear to be in the same universality class as connectivity percolation, suggesting superuniversality between all three types of transitions. Finally, the hierarchical lattice is analytically tractable and we find that the exponents depend both on the type of force and on the fraction of contacts at the Coulomb threshold. These combined results allow us to compare two universality classes on the same lattice via rigid clusters for the first time to highlight unifying and distinguishing concepts within the set of all possible rigidity transitions in disordered systems.
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