  In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries
all following an $a-$stable law divided by $N^{1/a}$. We prove that the least
singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the
Gaussian case, for almost all $a \in (0,2)$. This is proven by considering the
symmetrization of the matrix $D_{N}$ and using a version of the three step
strategy, a well known strategy in the random matrix theory literature. In
order to apply the three step strategy, we also prove an isotropic local law
for the symmetrization of matrices after slightly perturbing them by a Gaussian
matrix with a similar structure. The isotropic local law is proven for a
general class of matrices that satisfy some regularity assumption. We also
prove the complete delocalization for the left and right singular vectors of
$D_{N}$ at small energy, i.e., for energies at a small interval around $0$.
