The SUperluminous Supernova Host galaxIES (SUSHIES) survey aims to provide strong new constraints on the progenitors of superluminous supernovae (SLSNe) by understanding the relationship to their host galaxies. We present the photometric properties of 53 H-poor and 16 H-rich SLSN host galaxies out to z∼4. We model their spectral energy distributions to derive physical properties, which we compare with other galaxy populations. At low redshift, H-poor SLSNe are preferentially found in very blue, low-mass galaxies with high average specific star-formation rates. As redshift increases, the host population follows the general evolution of star-forming galaxies towards more luminous galaxies. After accounting for secular evolution, we find evidence for differential evolution in galaxy mass, but not in the B-band and the far UV luminosity (3σ confidence). Most remarkable is the scarcity of hosts with stellar masses above 1010 M⊙ for both classes of SLSNe. In the case of H-poor SLSNe, we attribute this to a stifled production efficiency above ∼0.4 solar metallicity. However, we argue that, in addition to low metallicity, a short-lived stellar population is also required to regulate the SLSN production. H-rich SLSNe are found in a very diverse population of star-forming galaxies. Still, the scarcity of massive hosts suggests a stifled production efficiency above ∼0.8 solar metallicity. The large dispersion of the H-rich SLSNe host properties is in stark contrast to those of gamma-ray burst, regular core-collapse SN, and H-poor SLSNe host galaxies. We propose that multiple progenitor channels give rise to this sub-class.