Magnetically tunable Feshbach resonances in ultracold atomic systems are chiefly identified and characterized through time-consuming atom loss spectroscopy. We describe an off-resonant dispersive optical probing technique to rapidly locate Feshbach resonances and demonstrate the method by locating four resonances of 87Rb, between the |F = 1,mF = 1〉 and |F = 2,mF = 0〉 states. Despite the loss features being ≲0.1 G wide, we require only 21 experimental runs to explore a magnetic field range >18 G, where 1G = 10-4 T. The resonances consist of two known s-wave features in the vicinity of 9 G and 18 G and two previously unreported p-wave features near 5G and 10 G. We further utilize the dispersive approach to directly characterize the two-body loss dynamics for each Feshbach resonance.