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Typical sensors detect small perturbations by measuring their effects on a physical observable, using a linear response principle (LRP). It turns out that once LRP is abandoned, new opportunities emerge. A prominent example is resonant systems operating near N th-order exceptional point degeneracies (EPDs) where a small perturbation ε ≪ 1 activates an inherent sublinear response ∼ ε N ≫ ε in resonant splitting. Here, we propose an alternative sublinear optomechanical sensing scheme that is rooted in Wigner’s cusp anomalies (WCAs), first discussed in the framework of nuclear reactions: a frequency-dependent square-root singularity of the differential scattering cross section around the energy threshold of a newly opened channel, which we use to amplify small perturbations. WCA hypersensitivity can be applied in a variety of sensing applications, besides optomechanical accelerometry discussed in this paper. Our WCA platforms are compact, do not require a judicious arrangement of active elements (unlike EPD platforms), and, if chosen, can be cavity free.