As part of the Virgo Cluster Survey Tracing Ionised Gas Emission (VESTIGE), a blind narrow-band Hα+ [NII] imaging survey of the Virgo cluster carried out with MegaCam at the CFHT, we discovered eight massive (10 10 M star 10 11 M ) lenticular galaxies with prominent ionised gas emission features in their inner (few kpc) regions. These features are either ionised gas filaments similar to those observed in cooling flows (2 galaxies), or thin discs with sizes 0.7 R(Hα) 2.0 kpc (6 galaxies), thus significantly smaller than those of the stellar disc (R(Hα) 7-22 % R iso (r)). These discs have morphological properties similar to those of the dust seen in absorption in high-resolution HST images. Using a unique set of multifrequency data, including new or archival ASTROSAT/UVIT, GALEX, HST, CFHT, Spitzer, and Herschel imaging data, combined with IFU (MUSE, ALMA) and long-slit (SOAR) spectroscopy, we show that while the gas located within these inner discs is photoionised by young stars, signaling ongoing star formation, the gas in the filamentary structures is shock-ionised. These discs have a star formation surface brightness similar to those observed in late-type galaxies. Because of their reduced size, however, these lenticular galaxies are located below the main sequence of unperturbed or cluster star-forming systems. By comparing the dust masses measured from absorption maps in optical images, from the Balmer decrement, or estimated by fitting the UV-to-far-IR spectral energy distribution of the target galaxies, we confirm that those derived from optical attenuation maps are heavily underestimated because of geometrical effects due to the relative distribution of the absorbing dust and the emitting stars. We have also shown that these galaxies have gas-to-dust ratios of G/D 80 320 30 , and that the star formation within these discs follows the Schmidt relation, albeit with an efficiency reduced by a factor of ∼ 2.5. Using our unique set of multifrequency data, we discuss the possible origin of the ionised gas in these objects, which suggests multiple and complex formation scenarios for massive lenticular galaxies in clusters.