We present new results from the SCUBA Local Universe Galaxy Survey (SLUGS), the first large systematic submillimetre survey of the local Universe. Since our initial survey of a sample of 104 IRAS-selected galaxies we have now completed a survey of a sample of 81 optically-selected galaxies, observed with the SCUBA camera on the James Clerk Maxwell Telescope. Since SCUBA is sensitive to the 90% of dust too cold to radiate significantly in the IRAS bands our new sample represents the first unbiased survey of dust in galaxies along the whole length of the Hubble sequence. We find little change in the properties of dust in galaxies along the Hubble sequence and detected 6 out of 11 elliptical galaxies. As in our earlier work on IRAS galaxies we find that the IRAS and submm fluxes are well-fitted by a two-component dust model with dust emissivity index beta=2. The major difference from our earlier work is that we find the ratio of the mass of cold dust to the mass of warm dust is much higher for our optically-selected galaxies and can reach values of ~1000. Comparison of the results for the IRAS- and optically-selected samples shows that there is a population of galaxies containing a large proportion of cold dust that is unrepresented in the IRAS sample. We derive local submm luminosity and dust mass functions, both directly from our optically-selected SLUGS sample, and by extrapolation from the IRAS PSCz survey using the method of Serjeant & Harrison, and find excellent agreement between the two. We find them to be well-fitted by Schechter functions except at the highest luminosities. We find that as a consequence of the omission of cold galaxies from the IRAS sample the luminosity function presented in our earlier work is too low by a factor of 2.