Contrary to received wisdom, conventional superconductors out-of equilibrium (NECSs) can be good ferromagnets and also excellent spin-dependent thermoelectric materials, and the superconducting wavefunction can be composed of spin-aligned pairs of electrons. My recent experiments demonstrating out-of-equilibrium magnetism, spin-charge separation and quasiparticle spin resonance in superconducting aluminium revealed that the magnetic properties of NECSs depend strongly on voltage- and magnetic-field-induced spin/charge currents and can thus be finely tuned. This work has motivated theoretical studies of the thermoelectric properties of NECSs, which are expected to show similar sensitivity to accessible experimental parameters. Thus, out-of-equilibrium spin currents and distributions in superconductors promise enhanced versatility and novel concepts for spin- and heat-based electronics – respectively spintronics and caloritronics – in particular through spin-sensitive distribution-function engineering. Building on my previous work, I propose to demonstrate, and establish techniques to finely control, the inter-conversion of non-equilibrium spin currents with both heat and charge currents in superconductor and topological-insulator-based devices. In doing so, I shall close the heat-spin-charge current triangle of mutual conversion and control in superconductors. This will in turn bring new techniques to bear on fundamental, long-standing problems in superconductivity, including the mechanisms of spin relaxation and decoherence; and the manipulation of the constituent electron spins in Cooper pairs to generate triplet superconductivity or spin-filtered, spatially-separated entangled electrons.