  This paper establishes that $N$-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.
