We discuss the instabilities upon the introduction of local Coulomb repulsion and explicit SU(N)-symmetric Heisenberg-like spin exchange to Dirac Fermi-surfaces, quadratic band touching point-like Fermi-surfaces and full Fermi-surfaces. The extension to higher symmetries allows us to study the melting of phases as a function of correlations as well as symmetry. We explore the quantum phases emerging in the SU(N)-symmetric Hubbard and Heisenberg models on the honeycomb lattice, Bernal-stacked honeycomb bilayer and the square lattice at half-filling by means of projective (zero temperature) quantum Monte Carlo simulations from SU(2) to the large-N limit for even values of N. We find SU(N)-spin symmetry broken phases at small values of N and weak coupling, and staggered flux phases, resonating valence bond plaquettes as well as columnar valence bond crystals in the limits of large-N and strong interactions.