We investigate the potential collider signatures of singly-charged and doubly-charged Higgs bosons such as those arising in minimal left-right symmetric models. Focusing on multileptonic probes in the context of the high-luminosity run of the Large Hadron Collider, we separately assess the advantages of the four-leptonic and trileptonic final states for a representative benchmark setup designed by considering a large set of experimental constraints. Our study establishes possibilities of identifying singly-charged and doubly-charged scalars at the Large Hadron Collider with a large significance, for luminosity goals expected to be reached during the high-luminosity phase of the Large Hadron Collider. We generalise our results and demonstrate that existing limits can in principle be pushed much further in the heavy mass regime.