Let $f: M \rightarrow N$ and $q \in N$ be a regular value, then $f^{-1}(q)$ is a submanifold of $M$.
Now assume that $q \in N$ is not a regular value, but you pick $K:=f^{-1}(q) \cap \{p \in M; Df|_p \text{ is surjective.}\}.$ Does this mean that $K$ is a manifold? Or more generally, is there a way out to define a manifold if our $q$ is not a regular value?
The set $U = \{x\in M : Df|_x\text{ is surjective}\}$ is open in $M$. Thus it is a manifold. If you restrict $f$ to $U$, then
$$K = (f|_U)^{-1}(q)$$
Thus $K$ is a manifold and is a submanifold in $U$.