We investigate a few scenaios that could lead to potential singularities in the 3D incompressible Euler equations. Of particular interest is the one that gives rise to a tornado like singularity. However, such a singular behavior may not be stable dynamically. There seems to be tremendous cancellation near the symmetry axis (near the center of the ``tornado''). Another potential singular behavior develops on a symmetry plane near the wall of an axi-symmetric cylinder. The solution seems to develop a self-similar singularity in the r-z plane. Using a very delicate method of analysis which involves computer aided proof, we prove the existence of a discrete family of self-similar profiles for a 1D model along the wall. The singularity exponent is determined by solving a nonlinear eigenvalue problem. Moreover, the self-similar profile seems to enjoy some stability property. The self-similar profiles we construct are non-conventional in the sense that they do not decay to zero at infinity but grow with certain fractional power. Such behavior is also observed in the numerical computation of the 3D Euler equations and is very different from the Leray type of self-similar solutions of the 3D Euler equations. This is a joint work with Dr. Guo Luo and Pengfei Liu.