Discrete Pin structures It is clear that an oriented manifold $M^n$ (with dimension $n$) admits spin structures if and only if its second Stiefel-Whitney class $[w^2]\in H^2(M,\mathbb Z_2)$ vanishes. In the construction of the simplicial complex upon triangulation of $M$, one can also use the $(n-2)$-th Stiefel-Whitney homology class $[w_{n-2}]$, which is the Poincare dual of $[w^2]$.
There is a relation between the discrete spin structure and Kasteleyn orientations, given in Kasteleyn (Ref 1).
For a spatial 2-manifold $M^2$ ($n=2$) with triangulation $T$, the Stiefel-Whitney homology class $[w_0]$ has a representative that is the summation of all vertices $v$ with some (mod 2) coefficients as follows Goldstein- Turner (Ref 2):
$$
w_0 = \sum_{v\in T} \# \{\sigma | v \subseteq \sigma \text{ is regular} \} \cdot v.
$$
Here, $v \subseteq \sigma$ means that $v$ is a sub-simplex of simplex $\sigma$. The subsimplex $v \subseteq \sigma$ is called regular if $v$ and $\sigma$ satisfy the certain relative positions. The $\# \{\sigma | v \subseteq \sigma \text{ is regular} \} \cdot v$ denotes the formal product of the (mod 2) number of regular pairs $v \subseteq \sigma$ and the vertex $v$.
One can call vertex $v$ singular if $\# \{\sigma | v \subseteq \sigma \text{ is regular} \}$ is odd. Then $w_0$ is the formal summation of all singular vertices. $w_0$ is a vector (0-th singular chain) in the vector space (of 0-th singular chains) spanned by the formal bases of all vertices with $\mathbb Z_2$ coefficients.
Note any 2D oriented manifold allow spin structures.

Question:
(1) Are there analogous discrete Pin structures (say Pin$^+$ or Pin$^-$) that we can define as "Stiefel-Whitney homology classes" for non-orientable $n$-manifolds $M$. Say $n=3, 4, 5$?
where Pin structure is given by:
$$
1\to \mathbb{Z}_2 \to \text{Pin}^{\pm}(n) \to \text{Spin}(n) \to 1
$$

In terms of cohomology class, for Pin$^+$, $w^2(M)=0$; and for Pin$^-$, $w^2(M)+(w^1)^2(M)=0$. (Here the usual notation shall be, for Pin$^+$, $w_2(M)=0$; and for Pin$^-$, $w_2(M)+w_1^2(M)=0$.
)

(2) What is the counterpart of Kasteleyn orientations, in
$$\text{Kasteleyn orientations v.s. discrete Spin structures}$$
$$\simeq \text{??? v.s. discrete Pin structures?}$$

References are welcome.

*

*P. W. Kasteleyn, Dimer Statistics and Phase Transitions Journal of Mathematical Physics 4, 287 (1963); https://doi.org/10.1063/1.1703953


*Richard Z Goldstein and Edward C Turner, “A formula for Stiefel-Whitney homology classes,” Proceedings of the American
Mathematical Society 58, 339–339 (1976).
see also:


*David Cimasoni and Nicolai Reshetikhin, “Dimers on Surface Graphs and Spin Structures.  i,” Communications in Mathematical
Physics 275, 187–208 (2007).


*David Cimasoni and Nicolai Reshetikhin, “Dimers on Surface Graphs and Spin Structures. II,” Communications in Mathematical
Physics 281, 445–468 (2008).
 A: Here are some partial answers.
For your first question: there are combinatorial formulas for all Stiefel-Whitney homology classes $w_k$, due to
Whitney and rediscovered by Cheeger.
Specifically, on a manifold with a triangulation $\Pi$, $w_k$ is represented by the sum of all $k$-simplices in
the barycentric subdivision of $\Pi$.
So the story you have for spin 2-manifolds and $w_0$ should generalize for a pin$+$ $n$-manifold,
where you want a trivialization of $[w_{n-2}]$, and can represent it using Whitney's combinatorial formula. For a
pin$-$ $n$-manifold, though, you'd need a trivialization of the Poincaré dual of $(w^1)^2$, which is not
calculated by Whitney's formula. It may be possible to imitate Whitney's argument for $(w^1)^2$, but I don't think it's
trivial to do so.
For your second question: in “Dimers on graphs in non-orientable
surfaces”, §§4–5, David Cimasoni describes a generalization of
Kastelyn orientations to pin$-$ surfaces, and proves in Theorem 5.3 that given a cell decomposition
and a dimer configuration on a closed surface $\Sigma$, generalized Kastelyn orientations are equivalent to
pin$-$-structures on $\Sigma$.
As far as I know, however, nothing has been written generalizing this past dimension 2, nor about combinatorial
pin$+$-structures in any dimension.
This question is related to another MathOverflow question from a few
years ago.
