  We present the dust properties and star-formation histories of local
submillimetre-selected galaxies in Herschel-ATLAS, classified by optical
morphology. 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.
  Comparing H-ATLAS galaxies to a control sample of optically selected
galaxies, we find 5.5% of luminous ETGs are detected in H-ATLAS. The H-ATLAS
ETGs contain a significant mass of cold dust: the mean dust mass is 5.5x10^7
Msun, with individual galaxies ranging from 9x10^5-4x10^8 Msun. This is
comparable to that of spirals in our sample, and is an order of magnitude more
dust than that found for the control ETGs, which have a median dust mass
inferred from stacking of (0.8-4.0)x10^6 Msun. The ETGs detected in H-ATLAS
have bluer NUV-r colours, higher specific star-formation rates and younger
stellar populations than ETGs which are optically selected, and may be
transitioning from the blue cloud to the red sequence. We also find that
H-ATLAS and control ETGs inhabit similar low-density environments. 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 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. 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. (Abridged)
