  We consider the focusing nonlinear Schr\"odinger equations $i\partial_t
u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$
supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and
$0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of
a self similar finite time blow up dynamics and provide a qualitative
description of the singularity formation near the blow up time
