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 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 -ATLAS Science Demonstration Phase data . Comparing to a control sample of optically selected galaxies , we find 5.5 % of luminous early-type galaxies are detected in H-ATLAS . The H-ATLAS early-type galaxies contain a significant mass of cold dust : the mean dust mass is 5.5 \times { 10 } ^ { 7 } M _ { \odot } , with individual galaxies ranging from 9 \times 10 ^ { 5 } -4 \times 10 ^ { 8 } M _ { \odot } . 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 ) \times 10 ^ { 6 } { M } _ { \odot } 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 can not be solely from stellar sources , and a large contribution from dust formed in the ISM 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 SSFR < 10 ^ { -11 } { yr ^ { -1 } } . We find these passive spirals have lower dust-to-stellar mass ratios , higher stellar masses and older stellar population ages than normal spirals . The passive spirals inhabit low density environments similar to those of the normal spiral galaxies in our sample . This shows that the processes which turn spirals passive do not occur solely in the intermediate density environments of group and cluster outskirts .