Superconductivity mediated by phonons is typically conventional, exhibiting a momentum-independent $s$-wave pairing function, due to the isotropic interactions between electrons and phonons along different crystalline directions. Here, by performing inelastic neutron scattering measurements on a superconducting single crystal of Sr$_{0.1}$Bi$_{2}$Se$_{3}$, a prime candidate for realizing topological superconductivity by doping the topo-logical insulator Bi$_{2}$Se$_{3}$, we found that there ex-ist singular phonons, with the linewidths of the acoustic phonons increasing substantially at long wavelengths, but only for those along the 001 direction. This observation indicates a large and singular electron-phonon coupling at small momenta, which we propose to give rise to the exotic $p$-wave nematic superconducting pairing in the MxBi$_{2}$Se$_{3}$ (M=Cu, Sr, Nb) superconductor family. Therefore, we show that these superconductors may be the first examples where electron-phonon interaction can induce more exotic superconducting pairing than the $s$-wave.