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 Painlevé 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.