  We study the $J/\psi \to \phi \pi^+ a_0(980)^- (a_0^- \to \pi^- \eta)$ decay,
evaluating the double mass distribution in terms of the $\pi^- \eta$ and $\pi^+
a^-_0$ invariant masses. We show that the $\pi^- \eta$ mass distribution
exhibits the typical cusp structure of the $a_0(980)$ seen in recent high
statistics experiments, and the $\pi^+ a^-_0$ spectrum shows clearly a peak
around $M_{\rm inv}(\pi^+ a^-_0)=1420 \,{\rm MeV}$, corresponding to a triangle
singularity. When integrating over the two invariant masses we find a branching
ratio for this decay of the order of $10^{-5}$, which is easily accessible in
present laboratories. We also call attention to the fact that the signal
obtained is compatible with a bump experimentally observed in the $\eta
\pi^+\pi^-$ mass distribution in the $J/\psi \to \phi \eta \pi^+\pi^-$ decay
and encourage further analysis to extract from there the $\phi \pi^+ a_0^-$ and
$\phi \pi^- a_0^+$ decay modes.
