We are motivated by practical applications of optical vortex lattices [1, 2] for interferometry [3, 4], which we have been developing using light optics. We have derived and experimentally demonstrated a quantitative form of ‘singularimetry’ [5], in which a 3-beam lattice of optical vortices provides an algebraic and localized method for measuring phase shifts imparted by an absorbing and refracting specimen in one arm of a 3-beam Mach-Zehnder interferometer [6]. For certain microscopy techniques, it may not be technologically feasible to use multiple beam splitters in this manner. For example, in electron holography, three beam electron vortex lattices have recently been experimentally realized, which could be used to map electric and magnetic fields on the nanoscale [7, 8]. In that context, restricting one electron beam to pass through a sample may pose significant engineering constraints. We have hence considered a different approach, in which all three beams pass through a sample of interest. To this end, we have developed an alternative differential form of singularimetry, which utilizes vortices and gradient singularities as topological fiducial markers in a structured-illumination context. This new approach provides a concise analytic measurement of phase gradients imparted by refracting specimens, yielding quantitative information that is both local and deterministic. We have accurately quantified our phase gradient experiments to demonstrate that lattices of wave field singularities can be used to detect subtle specimen variations with high precision.