If $f \colon M \to N$ is a smooth map between smooth manifolds, is it possible to find an open and dense set $M_0$, such that $f(M_0)$ is a manifold and $f \colon M_0 \to f(M_0)$ is a surjective submersion?

Edit: Since I think it isn't possible to always find such an open set, what assumptions do we need, to find such an open set?

More specifically, I have an hamiltonian Lie group action of $G$ on $M$ with $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$. In my paper the authors are assuming that $\Phi(M) = W$ is a manifold and $\Phi$ is a submersion on the image. But by the properties of the momentum map that means, that $\dim W = \dim G.x \quad \forall x \in M$. Since that means, that the $G$-orbits in $M$ should have all the same dimension, I think I have to restrict myself to the subset $M_0$ of maximal dimensional orbits. Then I would need that $f(M_0)$ is first a submanifold and second of dimension equal to the maximaldimensional $G$-orbits in $M$.