Two-dimensional fullerene networks have been synthesized in several forms, and it is unknown which monolayer form is stable under ambient conditions. Using first-principles calculations, I show that the believed stability of the quasi-tetragonal phases is challenged by mechanical, dynamic, or thermodynamic stability. For all temperatures, the quasi-hexagonal phase is thermodynamically the least stable. However, the relatively high dynamic and mechanical stabilities suggest that the quasi-hexagonal phase is intrinsically stronger than the other phases under various strains. The origin of the high stability and strength of the quasi-hexagonal phase can be attributed to the strong covalent C-C bonds that strongly hold the linked C 60 clusters together, enabling the closely packed hexagonal network. These results rationalize the experimental observations that so far only the quasi-hexagonal phase has been exfoliated experimentally as monolayers.