This thesis details recent results concerning the regularity, well-posedness, and long-time behavior of the first-order mean field games system with a local coupling. First, we prove that when the coupling is unbounded from below (the so-called blow-up assumption) and the density is positive, the system has classical solutions. Our starting point is a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions. Next, we investigate the extent to which these assumptions can be weakened in the one-dimensional setting. We first prove that the blow-up assumption can be removed, while still obtaining classical solutions, whose long-time behavior can then be fully characterized. Additionally, we show that, for interior times, the solution is still smooth if the positivity assumption of the density is weakened to an appropriate almost-everywhere positivity condition. Finally, we develop the more challenging setting where the density may vanish on a set of positive measure, which involves studying not only the regularity of the (weak) solutions but also of the emerging free boundary. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys $C^{1,1}$ regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be H\"older continuous up to the free boundary. The methods are based on the analysis of a new elliptic equation satisfied by the Lagrangian flow.