Orbit projections as fibrations

The orbit projection $\pi : M \to M/G$ of a proper $G$-manifold $M$ is a fibration if and only if all points in $M$ are regular. Under additional assumptions we show that $\pi$ is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: $\pi$ is a $G$-quasifibration if and only if all points are regular.