  We study analytically the Wigner function $W_N({\bf x},{\bf p})$ of $N$
noninteracting fermions trapped in a smooth confining potential $V({\bf x})$ in
$d$ dimensions. At zero temperature, $W_N({\bf x},{\bf p})$ is constant over a
finite support in the phase space $({\bf x},{\bf p})$ and vanishes outside.
Near the edge of this support, we find a universal scaling behavior of
$W_N({\bf x},{\bf p})$ for large $N$. The associated scaling function is
independent of the precise shape of the potential as well as the spatial
dimension $d$. We further generalize our results to finite temperature $T>0$.
We show that there exists a low temperature regime $T \sim e_N/b$ where $e_N$
is an energy scale that depends on $N$ and the confining potential $V({\bf
x})$, where the Wigner function at the edge again takes a universal scaling
form with a $b$-dependent scaling function. This temperature dependent scaling
function is also independent of the potential as well as the dimension $d$. Our
results generalize to any $d\geq 1$ and $T \geq 0$ the $d=1$ and $T=0$ results
obtained by Bettelheim and Wiegman [Phys. Rev. B ${\bf 84}$, 085102 (2011)].
