Aims. The general consensus is that late-type galaxies undergo intense star-formation, activity while early-type galaxies are mostly inactive. We question this general rule and investigate the existence of star-forming early-type and quiescent late-type galaxies in the local Universe. By computing the physical properties of these galaxies and by using information on their structural properties as well as the density of their local environment, we seek to understand the differences from their 'typical' counterparts.Methods. We made use of the multi-wavelength photometric data (from the ultraviolet to the sub-millimetre), for 2209 morphologically classified galaxies in the Galaxy And Mass Assembly survey. Furthermore, we separated the galaxies into subsets of star-forming and quiescent based on their dominant ionising process, making use of established criteria based on the W-H alpha width and the [N-II/H alpha] ratio. Taking advantage of the spectral energy distribution fitting code CIGALE, we derived galaxy properties, such as the stellar mass, dust mass, and star-formation rate, and we also estimated the unattenuated and the dust-absorbed stellar emission, for both the young ( 200 Myr) stellar populations.Results. We find that about 47% of E/S0 galaxies in our sample show ongoing star-formation activity and 8% of late-type galaxies are quiescent. The star-forming elliptical galaxies, together with the little blue spheroids, constitute a population that follows the star-forming main sequence of spiral galaxies very well. The fraction of the luminosity originating from young stars in the star-forming early-type galaxies is quite substantial (similar to 25%) and similar to that of the star-forming late-type galaxies. The stellar luminosity absorbed by the dust (and used to heat the dust grains) is highest in star-forming E/S0 galaxies (an average of 35%) followed by star-forming Sa-Scd galaxies (27%) with this fraction becoming significantly smaller for their quiescent analogues (6% and 16%, for E/S0 and Sa-Scd, respectively). Star-forming and quiescent E/S0 galaxies donate quite different fractions of their young stellar luminosities to heat up the dust grains (74% and 36%, respectively), while these fractions are very similar for star-forming and quiescent Sa-Scd galaxies (59% and 60%, respectively). Investigating possible differences between star-forming and quiescent galaxies, we find that the intrinsic (unattenuated) shape of the SED of the star-forming galaxies is, on average, very similar for all morphological types. Concerning their structural parameters, quiescent galaxies tend to show larger values of the r-band Sersic index and larger effective radii (compared to star-forming galaxies). Finally, we find that star-forming galaxies preferably reside in lower density environments compared to the quiescent ones, which exhibit a higher percentage of sources being members of groups.