Singularity of the potential function makes quantum tunneling, the case 0 () / V x V x of which is the subject of this article, mathematically underdetermined. To circumvent the difficulties it introduced in physics, a potential singularity cutoff is often used, followed by a reverse limit transition, or is a suitable self-adjoint extension of the Hamiltonian along the entire coordinate axis made. However, both of them somehow affect the singular nature of the problem, and so I discuss here how quantum tunneling will behave if the original singular nature of the Schrodinger equation left untouched. To do this, I use the property of the probability density current that the singularities are mutually destroyed in it. It is found that the mildly singular potential with 01 has a finite, but unusual tunnelling transparency, in particular, a non-zero value at zero energy of the incident particle. Тhe tunneling of 1D Coulomb potential (1 ) exhibits infinitely fast and complete oscillation at the zero energy boundary and a suppression to zero in the highenergy limit. In the more singular region with 1 , the tunneling becomes forbidden, thereby repeating the well-known result of the regularized counterparts.