We combine Herschel/SPIRE submillimetre (submm) observations with existing multiwavelength data to investigate the characteristics of low-redshift, optically red galaxies detected in submm bands. We select a sample of galaxies in the redshift range 0.01 <= z <= 0.2, having > 5s detections in the SPIRE 250 mu m submm waveband. Sources are then divided into two sub-samples of red and blue galaxies, based on their UV-optical colours. Galaxies in the red sample account for approximate to 4.2 per cent of the total number of sources with stellar masses M* >= 1010 M-circle dot. Following visual classification of the red galaxies, we find that greater than or similar to 30 per cent of them are early-type galaxies and greater than or similar to 40 per cent are spirals. The colour of the red-spiral galaxies could be the result of their highly inclined orientation and/or a strong contribution of the old stellar population. It is found that irrespective of their morphological types, red and blue sources occupy environments with more or less similar densities (i.e. the Sigma(5) parameter). From the analysis of the spectral energy distributions of galaxies in our samples based on MAGPHYS, we find that galaxies in the red sample (of any morphological type) have dust masses similar to those in the blue sample (i.e. normal spiral/star-forming systems). However, in comparison to the red-spirals and in particular blue systems, red-ellipticals have lower mean dust-to-stellar mass ratios. Besides galaxies in the red-elliptical sample have much lower mean star formation/specific star formation rates in contrast to their counterparts in the blue sample. Our results support a scenario where dust in early-type systems is likely to be of an external origin.