  Singularities that symbolize abrupt changes and exhibit extraordinary
behavior are of broad interest. We experimentally study optomechanically
induced singularities in a compound system consisting of a three-dimensional
aluminum superconducting cavity and a metalized high-coherence silicon nitride
membrane resonator. Mechanically-induced coherent perfect absorption and
anti-lasing occur simultaneously under a critical optomechanical coupling
strength. Meanwhile, the phase around the cavity resonance undergoes an abrupt
$\pi$-phase transition, which further flips the phase slope in the frequency
dependence. The observed infinite-discontinuity in the phase slope defines a
singularity, at which the group velocity is dramatically changed. Around the
singularity, an abrupt transition from an infinite group advance to delay is
demonstrated by measuring a Gaussian-shaped waveform propagating. Our
experiment may broaden the scope of realizing extremely long group delays by
taking advantage of singularities.
