The "VISTA near-infrared YJK_s_ survey of the Magellanic Clouds System" (VMC) is collecting deep K_s_-band time-series photometry of pulsating variable stars hosted by the two Magellanic Clouds and their connecting Bridge. In this paper, we present Y, J, K_s_ light curves for a sample of 4172 Small Magellanic Cloud (SMC) Classical Cepheids (CCs). These data, complemented with literature V values, allowed us to construct a variety of period-luminosity (PL), period-luminosity-color (PLC), and period-Wesenheit (PW) relationships, which are valid for Fundamental (F), First Overtone (FO), and Second Overtone (SO) pulsators. The relations involving the V, J, K_s_ bands are in agreement with their counterparts in the literature. As for the Y band, to our knowledge, we present the first CC PL, PW, and PLC relations ever derived using this filter. We also present the first near-infrared PL, PW, and PLC relations for SO pulsators to date. We used PW(V, K_s_) to estimate the relative SMC-LMC distance and, in turn, the absolute distance to the SMC. For the former quantity, we find a value of {Delta}{mu}=0.55+/-0.04mag, which is in rather good agreement with other evaluations based on CCs, but significantly larger than the results obtained from older population II distance indicators. This discrepancy might be due to the different geometric distributions of young and old tracers in both Clouds. As for the absolute distance to the SMC, our best estimates are {mu}_SMC_=19.01+/-0.05mag and {mu}_SMC_=19.04+/-0.06mag, based on two distance measurements to the LMC which rely on accurate CC and eclipsing Cepheid binary data, respectively.