Abstract The VISTA near-infrared YJK s survey of the Magellanic System (VMC) is collecting deep K s-band time-series photometry of pulsating stars hosted by the two Magellanic Clouds and their connecting bridge. Here, we present Y, J, K s light curves for a sample of 717 Small Magellanic Cloud (SMC) Classical Cepheids (CCs). These data, complemented with our previous results and V magnitude from literature, allowed us to construct a variety of period–luminosity and period–Wesenheit relationships, valid for Fundamental, First and Second Overtone pulsators. These relations provide accurate individual distances to CCs in the SMC over an area of more than 40 deg2. Adopting literature relations, we estimated ages and metallicities for the majority of the investigated pulsators, finding that (i) the age distribution is bimodal, with two peaks at 120 ± 10 and 220 ± 10 Myr; (i) the more metal-rich CCs appear to be located closer to the centre of the galaxy. Our results show that the three-dimensional distribution of the CCs in the SMC is not planar but heavily elongated for more than 25–30 kpc approximately in the east/north-east towards south-west direction. The young and old CCs in the SMC show a different geometric distribution. Our data support the current theoretical scenario predicting a close encounter or a direct collision between the Clouds some 200 Myr ago and confirm the presence of a Counter-Bridge predicted by some models. The high-precision three-dimensional distribution of young stars presented in this paper provides a new test bed for future models exploring the formation and evolution of the Magellanic System.