The following is a result from Shub's monograph "Global Stability of Dynamical Systems".
I dabble in the proof, and it appears to me that the existence of $W^{\rm cu}\_{\rm loc}$ does not rely on the assumption that $Df(0)$ is invertible, while that of the existence of $W^{\rm cs}\_{\rm loc}$ does.


Question: Can $W^{\rm cs}\_{\rm loc}$ fail to exist if $Df(0)$ is singular?


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Theorem: Let $0$ be a fixed point for the $C^r$
local diffeomorphism $f: U \rightarrow R^n$ where $U$ is a neighborhood of zero in $R^n$ and
$\infty > r \geq 1$. Let $E^{\rm s} \oplus E^{\rm c} \oplus E^{\rm u}$ be the invariant splitting of $R^n$ into the generalized
eigenspaces of $Df(0)$ corresponding to eigenvalues of absolute value less
than one, equal to one, and greater than one. To each of the five $Df(0)$ invariant
subspaces $E^{\rm s}$, $E^{\rm s} \oplus E^{\rm c}$, $E^{\rm c}$, $E^{\rm c} \oplus E^{\rm u}$, and $E^{\rm u}$ there is associated a local f invariant $C^r$
embedded disc $W^{\rm s}\_{\rm loc}$, $W^{\rm cs}\_{\rm loc}$, $W^{\rm c}\_{\rm loc}$, $W^{\rm cu}\_{\rm loc}$, $W^{\rm u}\_{\rm loc}$
tangent to the linear subspace at $0$ and a ball $B$ around zero in a (suitably defined) norm such that:


1. $W^{\rm s}\_{\rm loc} = \{ x\in B | \mbox{ $f^n(x) \in B$ for all $n\geq 0$ and $d(f^n(x),0)$ tends to zero exponentially} \}$.
$f: W^{\rm s}\_{\rm loc} \rightarrow W^{\rm s}\_{\rm loc}$ is a contraction mapping.
2. $f(W^{\rm cs}\_{\rm loc}) \cap B \subset W^{\rm cs}\_{\rm loc}$. If $f^n(x) \in B$ for all $n \geq 0$, then $x \in W^{\rm cs}\_{\rm loc}$.
3. $f(W^{\rm c}\_{\rm loc}) \cap B \subset W^{\rm c}\_{\rm loc}$. If $f^n(x) \in B$ for all $n \in Z$, then $x \in W^{\rm c}\_{\rm loc}$.
4. $f(W^{\rm cu}\_{\rm loc}) \cap B \subset W^{\rm cu}\_{\rm loc}$. If $f^n(x) \in B$ for all $n \leq 0$, then $x \in W^{\rm cu}\_{\rm loc}$.
5. $W^{\rm u}\_{\rm loc} = \{ x\in B | \mbox{ $f^n(x) \in B$ for all $n \leq 0$ and $d(f^n(x),0)$ tends to zero exponentially} \}$. $f^{-1}: W^{\rm u}\_{\rm loc} \rightarrow W^{\rm u}\_{\rm loc}$ is a contraction mapping.



Existence of center-stable manifold when the Jacobian is singular?
For both manifolds we do not need the strong invertibility. Moreover, for $W^{cs}$ this is stated in Exercise III.2, p.68 from the mentioned monograph. However, I will give below a more geometrical view on the problem.


In fact, to construct $W^{ss}$ (strongly stable), $W^{su}$ (strongly unstable), $W^{cs}$ (center-stable), $W^{cu}$ (center-unstable) and $W^{c}$ (center) manifolds, it is only required a proper trichotomy, which requires an invertibility of $Df$ on the center-unstable part of the linearized system (see below). Note also that the "center submanifold" is better to understand as the submanifold with uncertain behavior. For many authors the center manifold exists only if there are eigenvalues on the unit circle (or on the imaginary axis for ODEs). This is a kind of nonsense since in practice we study systems depending on parameters by reducing them to the center manifold. Eigenvalues on the unit circle exist only for critical parameters and by such a strange definition there is no center manifold at any neighborhood of critical parameters. But the submanifolds tangent to the corresponding generalized eigenspace (with eigenvalues close to the unit circle) exist and the same authors also call them center manifolds that disagrees with the initial definition.


Let me give a brief geometric picture that explains the assumptions. Firstly, I assume that $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ a $C^{1}$-mapping (not necessarily a "local diffeomorphism") such that $f(0)=0$ and $f(x)=(D\_{0}f)x + h(x)$ with $\|D\_{p}h\| \leq \varepsilon$ at any point $p \in \mathbb{R}^{n}$ and sufficiently small $\varepsilon>0$. Such a situation can be achieved from any general smooth mapping (satisfying $f(0)=0$) after a proper truncation outside a small neighborhood of $0$.


Suppose that $0 < \frac{1}{\nu} < 1 < \frac{1}{\mu}$ (for convenience; in fact, we just need $1/\nu < 1/\mu$) and that the circles of radii $\frac{1}{\nu}$ and $\frac{1}{\mu}$ avoid the spectrum of $D\_{0}f$. Let $\mathbb{R}^{n} = \mathbb{E}^{ss} \oplus \mathbb{E}^{c} \oplus \mathbb{E}^{su}$ be the invariant splitting w.r.t. $D\_{0}f$, where $\mathbb{E}^{ss}$ corresponds to the eigenvalues with $|\lambda| < \frac{1}{\nu}$, $\mathbb{E}^{c}$ corresponds to the eigenvalues with $\frac{1}{\nu} < |\lambda| < \frac{1}{\mu}$ and $\mathbb{E}^{su}$ corresponds to the eigenvalues with $|\lambda| >\frac{1}{\mu}$.


Let us fix an inner product in $\mathbb{R}^{n}$ and the induced norm $|\cdot|$. Put $A:=D\_{0}f$ and define two quadratic forms $V(\cdot)$ and $W(\cdot)$ as
$$V(x) := \begin{cases}
\sum\_{k=0}^{\infty} \nu^{2k} |A^{k}x|^{2}, \text{ for } x \in \mathbb{E}^{ss}, \\
-\sum\_{-\infty}^{k=-1} \nu^{2k} |A^{k}x|^{2}, \text{ for } x \in \mathbb{E}^{c} \oplus \mathbb{E}^{su}
\end{cases}$$
and
$$W(x) := \begin{cases}
\sum\_{k=0}^{\infty} \mu^{2k} |A^{k}x|^{2}, \text{ for } x \in \mathbb{E}^{ss} \oplus \mathbb{E}^{c}, \\
-\sum\_{-\infty}^{k=-1} \mu^{2k} |A^{k}x|^{2}, \text{ for } x \in \mathbb{E}^{su}
\end{cases}$$
Here $A^{k}x$ for $k<0$ and $x \in \mathbb{E}^{c} \oplus \mathbb{E}^{su}$ means $(A^{-k})^{-1}x$ when $A$ is restricted to $\mathbb{E}^{c} \oplus \mathbb{E}^{su}$. *This is the only place where a proper trichotomy of $A$ is used*. It is clear that $V(\cdot)$ satisfies for any integer $N \geq 1$ and $x \in \mathbb{R}^{n}$ the relation
$$ \nu^{2N} V(A^{N} x) - V(x) = -\sum\_{k=0}^{N-1} \nu^{2k}|A^{k}x|^{2}.$$
Moreover, for $f$ given as above with a sufficiently small $\varepsilon>0$ we have the inequality (for any $x\_{1},x\_{2} \in \mathbb{R}^{n})$
$$ \nu^{2N} V(f^{N}(x\_{1})-f^{N}(x\_{2})) - V(x\_{1}-x\_{2}) \leq -\delta \sum\_{k=0}^{N-1} \nu^{2k} | f^{k}(x\_{1})-f^{k}(x\_{2}) |^{2}$$
with a sufficiently small $\delta>0$ (hint: it is sufficient to verify it for $N=1$). Analogous statements are satisfied for $W$.


It turns out that these conditions on $f$ w.r.t. $V(\cdot)$ and $W(\cdot)$ are sufficient to reconstruct all the submanifolds and foliations for $f$.


For example, consider two points $x\_{0}$ and $y\_{0}$ which admit negative orbits w.r.t. $f$, i.e. sequences $x\_{k}$ and $y\_{k}$ for $k=0,-1,-2,\ldots$ with $f(x\_{k})=x\_{k+1}$ and $f(y\_{k}) = y\_{k+1}$ for any $k=-1,-2,\ldots$. Such orbits are said to be negatively pseudo-ordered w.r.t. $V$ if $V(x\_{k}-y\_{k}) \leq 0$ for any $k=0,-1,-2,\ldots$. For a fixed negative orbit $\{x\_{k}\}$ let $[\{x\_{k}\}]$ be the class of all such $\{y\_{k}\}$ as above and let $[\{x\_{k}\}](0)$ be the set of all $y\_{0}$ taking over such $\{y\_{k}\}$. Then $W^{cu}(0)$ is defined as $[\{0\}](0)$.


Moreover, let us say that two points $x$ and $y$ are positively equivalent w.r.t. $W$ if $W(f^{k}(x)-f^{k}(y)) \geq 0$ for any $k=0,1,2,\ldots$. Positive equivalence is an equivalence relation. Let $[x]^{+}$ be the equivalence class of $x$. Then $W^{sc}(0)$ is given by $[0]^{+}$.


So, it is not required for $A=D\_{0}f$ (and especially $f$) to be everywhere invertible to construct the submanifolds and in finite dimensions $A$ is automatically invertible where it is required.


For the details I refer to my [preprint](https://arxiv.org/abs/2012.03821) (a bit outdated), where the continuous case is studied in the context of cocycles in Banach spaces.


