  We compute the joint statistics of the momenta $p_i$ of $N$ non-interacting
fermions in a trap, near the Fermi edge, with a particular focus on the largest
one $p_{\max}$. For a $1d$ harmonic trap, momenta and positions play a
symmetric role and hence, the joint statistics of momenta is identical to that
of the positions. In particular, $p_{\max}$, as $x_{\max}$, is distributed
according to the Tracy-Widom distribution. Here we show that novel "momentum
edge statistics" emerge when the curvature of the potential vanishes, i.e. for
"flat traps" near their minimum, with $V(x) \sim x^{2n}$ and $n>1$. These are
based on generalisations of the Airy kernel that we obtain explicitly. The
fluctuations of $p_{\max}$ are governed by new universal distributions
determined from the $n$-th member of the second Painlev\'e hierarchy of
non-linear differential equations, with connections to multicritical random
matrix models. Finite temperature extensions and possible experimental
signatures in cold atoms are discussed.
