Consider a smooth map $F\colon \mathbb{R}^n \to \mathbb{R}^n$. A basic and common type of singularity is the fold. Informally, along a hypersurface $S$ of critical points, a fold map glues the two sides of $S$ together and sends them to one side of the hypersurface $F(S)$.
Fold can be locally modeld by $F(u,v)=(u, v^2)$ where $u\in \mathbb{R}^{n-1}, v\in \mathbb{R}$.
Intuitvely, fold is tame. In fact, let $D=\{(u,v)\colon v\geq 0\}$. Then $F$ is a homeomorphism from $D$ to $D$ and a diffeomorphism from $\text{int}(D)$ to $\text{int}(D)$.
However, $F$ can never be a diffeomorphism from $D$ to $D$ as manifolds with boundaries.
In textbook terms, diffeomorphisms require the Jacobian induces isomorphisms between tangent spaces everywhere. Since the boundary of $D$ consists of singular points, $JF$ fails to do so.
This is also visible by looking at the behavior in the normal direction. Notice that for any one-dimensional map $f$, to achieve the folding while remaining smooth, $f’(0)$ must be zero. Thus, the inverse of $f$, if it exists, is not differentiable at $0$. In terms of $F$, this means that the normal component of the tangent space must collapse.