We investigate the self-similar singularity for a 1D model of the 3D axisymmetric Euler equations, which is motivated by a particular singularity formation scenario observed in a recent numerical computation. 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 this model and analyze their far-field properties. The self-similar profiles we obtain agree with those obtained by direct numerical simulations of the model. Moreover, the self-similar profile enjoys 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 of Luo and Hou, which is very different from the Leray type of self-similar solutions of the 3D Euler equations. We are currently investigating the possibility of extending this method to study the singularity formation of the 2D Boussinesq system. This is a joint work with Pengfei Liu.