We present analysis of the normalised 21-cm bispectrum from fully-numerical simulations of intergalactic-medium heating by stellar sources and high-mass X-ray binaries (HMXB) during the cosmic dawn. Lyman-$alpha$ coupling is assumed to be saturated, we therefore probe the nature of non-Gaussianities produced by X-ray heating processes. We find the evolution of the normalised bispectrum to be very different from that of the power spectrum. It exhibits a turnover whose peak moves from large to small scales with decreasing redshift, and corresponds to the typical separation of emission regions. This characteristic scale reduces as more and more regions move into emission with time. Ultimately, small-scale fluctuations within heated regions come to dominate the normalised bispectrum, which at the end of the simulation is almost entirely driven by fluctuations in the density field. To establish how generic the qualitative evolution of the normalised bispectrum we see in the stellar + HMXB simulation is, we examine several other simulations - two fully-numerical simulations that include QSO sources, and two with contrasting source properties produced with the semi-numerical simulation 21cmFAST. We find the qualitative evolution of the normalised bispectrum during X-ray heating to be generic, unless the sources of X-rays are, as with QSOs, less numerous and so exhibit more distinct isolated heated profiles. Assuming mitigation of foreground and instrumental effects are ultimately effective, we find that we should be sensitive to the normalised bispectrum during the epoch of heating, so long as the spin temperature has not saturated by $z approx 19$.
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