The spatial tunability of the electron density in graphene enables the dynamic engineering of metasurfaces in the form of conductivity gratings, which can bridge the momentum gap between incident radiation and surface plasmons. Here, we discuss singular graphene metasurfaces, whose conductivity is strongly suppressed at the grating valleys. By analytically characterising their plasmonic response via transformation optics, we first review the physical principles underlying these structures, which were recently found to exhibit broadband, tunable THz absorption. We characterise the spectrum with different common substrates and then move to study in further detail how conductivity gratings may be finely tuned by placing an array of charged gold nanowires at sub-micron distance from the graphene.