A survey is presented of the submillimetre continuum spectra from dust around 73 young stellar objects (YSOs) associated with Herbig-Haro objects and molecular outflows. Spectra at millimetre to submillimetre wavelengths taken with similar beamsizes have been obtained for 42 of the objects, and the remainder either have been detected or have upper limits at 800 mu m. The data are combined with far-infrared IRAS fluxes, and then compared with published submillimetre data from other types of YSO; this results in a data set of similar to 150 submillimetre spectra. We have then employed colour-colour diagrams to investigate the differences in submillimetre continuum spectra between classes of YSOs. The majority of targets are optically obscured YSOs (generally Lada Class I); all are found to have a submillimetre spectral index (alpha(mm)) of similar to 3.5. None of these objects has alpha(mm) less than or equal to 3.0. This is significantly different from published optically visible T Tauri stars (Class II-III), many of which have alpha(mm) similar to 2.5. The few main-sequence stars so far observed in the submillimetre generally also have a relatively low value of alpha(mm). We fmd that objects which might be classified as T Tauri stars, but which are not directly visible, have higher values of alpha(mm) Assuming isothermal optically thin dust emission, this implies a dust emissivity index beta of similar to 1.5 for embedded objects, and similar to 0.5 for visible objects. The reduced alpha(mm) in low optical extinction sources can be attributed to either grain growth or the formation of a compact optically thick circumstellar disc, and we discuss these two options in the light of the new data. If, as the results suggest, large fractal grains are present in the more evolved systems, then these will have higher millimetre mass opacity, implying that the determination of the circumstellar mass evolution from submillimetre continuum fluxes must take into account the evolution of the dust itself. High-mass embedded YSOs also have spectra very similar to their low-mass counterparts, but with higher dust temperature (again, under the isothermal optically thin assumption). Also, most Class 0 YSOs have high values of alpha(mm), although three have flatter spectra more typical of fractal dust, which is difficult to explain under simple grain coagulation models. The submillimetre continuum flux is also compared with the intensity of two molecular lines: (CO)-O-18 J = 2-1 and H2CO 3(03) - 2(02). The integrated emission from the higher density tracer, H2CO, is well correlated with continuum flux. The (CO)-O-18 line shows a less significant correlation, particularly at low masses, suggesting that this may not be a good molecule to search for low-mass YSOs. By comparing T Tauri stars and younger YSOs with the same continuum flux, we find evidence that T Tauri stars are unusually weak in (CO)-O-18 emission, and we give some possible explanations.