We combine SAURON integral field data of a representative sample of local early-type, red sequence galaxies with Spitzer/Infrared Array Camera imaging in order to investigate the presence of trace star formation in these systems. With the Spitzer data, we identify galaxies hosting low-level star formation, as traced by polycyclic aromatic hydrocarbon emission, with measured star formation rates that compare well to those estimated from other tracers. This star formation proceeds according to established scaling relations with molecular gas content, in surface density regimes characteristic of disc galaxies and circumnuclear starbursts. We find that star formation in early-type galaxies happens exclusively in fast-rotating systems and occurs in two distinct modes. In the first, star formation is a diffuse process, corresponding to widespread young stellar populations and high molecular gas content. The equal presence of co- and counter-rotating components in these systems strongly implies an external origin for the star-forming gas, and we argue that these star formation events may be the final stages of (mostly minor) mergers that build up the bulges of red sequence lenticulars. In the second mode of star formation, the process is concentrated into well-defined disc or ring morphologies, outside of which the host galaxies exhibit uniformly evolved stellar populations. This implies that these star formation events represent rejuvenations within previously quiescent stellar systems. Evidence for earlier star formation events similar to these in all fast-rotating early-type galaxies suggests that this mode of star formation may be common to all such galaxies, with a duty cycle of roughly 1/10, and likely contributes to the embedded, corotating inner stellar discs ubiquitous in this population.