Disc spectra from highly luminous black hole binaries are observed to be rather simple, despite theoretical predictions to the contrary. We collate the disc-dominated spectra from multiple observations of 10 separate sources and show that they are consistent with a simple multicolour blackbody with a disc luminosity of LdiscT4 over two orders of magnitude. This is probably compatible with the predictions of standard Shakura–Sunyaev α-disc theory. However, these models also predict that the radiation-pressure-dominated disc is unstable, in strong conflict to the observed lack of variability of the light curves. We show that this discrepancy is unlikely to be resolved by modifications such as additional energy loss in a jet, a wind or a corona. This motivates consideration of alternative disc models, in particular the β-discs, where the viscous heating is proportional to gas pressure, which are stable. While proper calculations have yet to be performed, it seems likely that such models predict significant departures from a simple LdiscT4 relation, again in conflict with the observations. More generally, this shows that accretion disc spectra can give reliable constraints on the disc structure, and should be used to test the next generation of accretion disc models, in which the disc viscosity is calculated self-consistently from the magnetically generated turbulent dynamo.