We use a large sample of upper limits and accurate estimates of supermassive black hole (SMBH) masses coupled with libraries of host galaxy velocity dispersions, rotational velocities and photometric parameters extracted from Sloan Digital Sky Survey i-band images to establish correlations between the SMBH and host galaxy parameters. We test whether the mass of the black hole, M•, is fundamentally driven by either local or global galaxy properties. We explore correlations between M• and stellar velocity dispersion σe, i-band bulge luminosity Li, bulge, bulge mass Mbulge, bulge Sérsic index n, bulge mean effective surface brightness 〈μe, bulge〉, i-band luminosity of the galaxy Li, gal, galaxy stellar mass , maximum circular velocity Vc, and galaxy dynamical and effective masses Mdyn, gal and Me, gal. We verify the tightness of the M•–σe relation and find that correlations with other galaxy parameters do not yield tighter trends. We do not find differences in the M•–σe relation of barred and unbarred galaxies. The M•–σe relation of pseudo-bulges is also coarser and has a different slope than that involving classical bulges. The M•–Mbulge is not as tight as the M•–σe relation, despite the bulge mass proving to be a better proxy of M• than bulge luminosity, and despite adding the bulge effective radius as an additional fitting parameter. Contrary to various published reports, we find a rather poor correlation between M• and n (or 〈μe, bulge〉), suggesting that M• is not related to the bulge light concentration. The correlations between M• and galaxy luminosity or mass are not a marked improvement over the M•–σe relation. These scaling relations depend sensitively on the host galaxy morphology: early-type galaxies follow a tighter relation than late-type galaxies. If Vc is a proxy for the dark matter halo mass, the large scatter of the M•–Vc relation then suggests that M• is more coupled to the baryonic rather than the dark matter. We have tested the need for a third parameter in the M• scaling relations, through various linear correlations with bulge and galaxy parameters, only to confirm that the Fundamental Plane of the SMBH is mainly driven by σe with a small tilt due to the effective radius. We provide a compendium of galaxy structural properties for most of the SMBH hosts known to date.