We use a complete, unbiased sample of 257 dwarf (10^8^M_{sun}_<Mstar<10^9.5^M_{sun}_) galaxies at z<0.08, in the COSMOS field, to study the morphological mix of the dwarf population in low-density environments. Visual inspection of extremely deep optical images and their unsharp-masked counterparts reveals three principal dwarf morphological classes. 43 per cent and 45 per cent of dwarfs exhibit the traditional 'early-type' (elliptical/S0) and 'late-type' (spiral) morphologies, respectively. However, 10 per cent populate a 'featureless' class, that lacks both the central light concentration seen in early-types and any spiral structure - this class is missing in the massive-galaxy regime. 14 per cent, 27 per cent, and 19 per cent of early-type, late-type, and featureless dwarfs respectively show evidence for interactions, which drive around 20 per cent of the overall star formation activity in the dwarf population. Compared to their massive counterparts, dwarf early-types show a much lower incidence of interactions, are significantly less concentrated and share similar rest-frame colours as dwarf late-types. This suggests that the formation histories of dwarf and massive early-types are different, with dwarf early-types being shaped less by interactions and more by secular processes. The lack of large groups or clusters in COSMOS at z<0.08, and the fact that our dwarf morphological classes show similar local density, suggests that featureless dwarfs in low-density environments are created via internal baryonic feedback, rather than by environmental processes. Finally, while interacting dwarfs can be identified using the asymmetry parameter, it is challenging to cleanly separate early and late-type dwarfs using traditional morphological parameters, such as 'CAS', M20, and the Gini coefficient (unlike in the massive-galaxy regime).