The Magellanic Clouds are nearby dwarf irregular galaxies whose morphologies show different properties when traced by different stellar populations, making them an important laboratory for studying galaxy morphologies. We study the morphology of the Magellanic Clouds using data from the VISTA survey of the Magellanic Clouds system (VMC). We used about 10 and 2.5 million sources across an area of ̃105 deg2 and ̃42 deg2 towards the Large and Small Magellanic Cloud (LMC and SMC), respectively. We estimated median ages of stellar populations occupying different regions of the near-infrared (J - Ks, Ks) colour-magnitude diagram. Morphological maps were produced and detailed features in the central regions were characterised for the first time with bins corresponding to a spatial resolution of 0.13 kpc (LMC) and 0.16 kpc (SMC). In the LMC, we find that main sequence stars show coherent structures that grow with age and trace the multiple spiral arms of the galaxy, star forming regions become dimmer as we progress in age, while supergiant stars are centrally concentrated. Intermediate-age stars, despite tracing a regular and symmetrical morphology, show central clumps and hints of spiral arms. In the SMC, young main sequence stars depict a broken bar. Intermediate-age populations show signatures of elongation towards the Magellanic Bridge that can be attributed to the LMC-SMC interaction ̃200 Myr ago. They also show irregular central features suggesting that the inner SMC has also been influenced by tidal interactions.