  We consider a random symmetric matrix ${\bf X} = [X_{jk}]_{j,k=1}^n$ with
upper triangular entries being i.i.d. random variables with mean zero and unit
variance. We additionally suppose that $\mathbb E |X_{11}|^{4 + \delta} =:
\mu_{4+\delta} < \infty$ for some $\delta > 0$. The aim of this paper is to
significantly extend recent result of the authors [18] and show that with high
probability the typical distance between the Stieltjes transform of the
empirical spectral distribution (ESD) of the matrix $n^{-\frac{1}{2}} {\bf X}$
and Wigner's semicircle law is of order $(nv)^{-1} \log n$, where $v$ denotes
the distance to the real line in the complex plane. We apply this result to the
rate of convergence of the ESD to the distribution function of the semicircle
law as well as to rigidity of eigenvalues and eigenvector delocalization
significantly extending a recent result by G\"otze, Naumov and Tikhomirov [19].
The result on delocalization is optimal by comparison with GOE ensembles.
Furthermore the techniques of this paper provide a new shorter proof for the
optimal $O(n^{-1})$ rate of convergence of the expected ESD to the semicircle
law.
