Motivated by the surge in research activities on graphene, we investigate instabilities of electrons on the honeycomb lattice, interacting by onsite and nearest-neighbor terms, using a renormalization group scheme. Near half band filling, critical minimal interaction strengths are required for instabilities toward antiferromagnetic or charge-density-wave order. Away from half-filling, f-wave triplet-pairing and d + id singlet-pairing instabilities are found to emerge out of density-wave regimes.