The encoding of SU(N) symmetric spin degrees of freedom in ultra-cold atom systems has recently brought the realization of antiferromagnetic order in optical lattices into close reach. We explore the quantum phases and phase transitions emerging in the most fundamental interacting models on the lattice, the SU(N)-symmetric Hubbard and Heisenberg model, by means of projective quantum Monte Carlo simulations from SU(2) to the large-N limit. We discuss the instabilities upon the introduction of local Coulomb repulsion and explicit SU(N)-symmetric Heisenberg-like spin exchange to different Fermi-surfaces. The extension to higher symmetries allows us to study the melting of phases as a function of correlations as well as symmetry.