We present the dust properties and star formation histories of local submillimetre-selected galaxies, classified by optical morphology. Most of the galaxies are late types and very few are early types. The early-type galaxies (ETGs) that are detected contain as much dust as typical spirals, and form a unique sample that has been blindly selected at submillimetre wavelengths. Additionally, we investigate the properties of the most passive, dusty spirals. We morphologically classify 1087 galaxies detected in the Herschel-Astrophysical Terahertz Large Area Survey (H-ATLAS) Science Demonstration Phase data. Comparing to a control sample of optically selected galaxies, we find 5.5 per cent of luminous ETGs are detected in H-ATLAS. The H-ATLAS ETGs contain a significant mass of cold dust: the mean dust mass is 5.5 x 10(7)M(circle dot), with individual galaxies ranging from 9 x 10(5) to 4 x 10(8)M(circle dot). This is comparable to that of spiral galaxies in our sample, and is an order of magnitude more dust than that found for the control early-types, which have a median dust mass inferred from stacking of (0.8-4.0) x 10(6)M(circle dot) for a cold dust temperature of 25-15 K. The early-types detected in H-ATLAS tend to have bluer NUV - r colours, higher specific star formation rates and younger stellar populations than early-types which are optically selected, and may be transitioning from the blue cloud to the red sequence. We also find that H-ATLAS and control early-types inhabit similar low-density environments. We investigate whether the observed dust in H-ATLAS early-types is from evolved stars, or has been acquired from external sources through interactions and mergers. We conclude that the dust in H-ATLAS and control ETGs cannot be solely from stellar sources, and a large contribution from dust formed in the interstellar medium or external sources is required. Alternatively, dust destruction may not be as efficient as predicted. We also explore the properties of the most passive spiral galaxies in our sample with specific star formation rate (SSFR)