This paper investigates the three-dimensional instabilities of the flow past a periodically heaving airfoil. By comparison with a pitching foil [Deng et al., Phys. Rev. E 92, 063013 (2015)PLEEE81539-375510.1103/PhysRevE.92.063013], here we present distinctive characteristics for the heaving foil, particulary regarding its Floquet modes. By increasing the frequency (Sr), or equivalently decreasing the amplitude (A_{D}) along the marginal stability curve in the (Sr,A_{D}) phase space, the critical Floquet mode emerges sequentially as A, quasiperiodic (QP), and B. It is interesting to note that both modes A and B are synchronous with the base flow, in contrast to the quasiperiodic mode QP. To further investigate the instability across the marginal curve, we fix the frequency at Sr=0.187, of which the critical Floquet mode is located in the synchronous regime, while varying A_{D} around the critical point. We find that the dominant mode switches from mode A to mode B, while mode QP never becomes critical as we increase A_{D}. We note that mode S, a subharmonic mode, can also be unstable, which, however, is not physically realizable, because the magnitude of its Floquet multiplier is always smaller than that of mode B. We have also studied the influence of various Reynolds numbers at the same critical point on the marginal stability curve, with the results resembling that by varying the amplitude A_{D}.