  This paper establishes that a class of $N$-player stochastic games with
singular controls, either of bounded velocity or of finite variation, can both
be approximated by mean field games (MFGs) with singular controls of bounded
velocity. More specifically, it shows (i) the optimal control to an MFG with
singular controls of a bounded velocity $\theta$ is shown to be an
$\epsilon_N$-NE to an $N$-player game with singular controls of the bounded
velocity, with $\epsilon_N = O(\frac{1}{\sqrt{N}})$, and (ii) the optimal
control to this MFG is an $(\epsilon_N + \epsilon_{\theta})$-NE to an
$N$-player game with singular controls of finite variation, where
$\epsilon_{\theta}$ is an error term that depends on $\theta$. This work
generalizes the classical result on approximation $N$-player games by MFGs, by
allowing for discontinuous controls.
