Can we 'build' spinor structure not only from a Riemann Manifold but 'extract it' also from another algebraic structures?

Question

I want to understand what type of structures are Spin Structure: are a monoids, ringoids, groups?
Can we build spinor structure find also from another structures not 'extract it' only from a Riemann Manifold?
Can you help me to understand what are structure and other ways to 'find' spinor structures

In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors. One method for dealing with this problem is to require that M has a spin structure.

1. This is not always possible since there is potentially a topological obstruction to the existence of spin structures.

2. Spin structures will exist if and only if the second Stiefel–Whitney class w2(M) ∈ H2(M, Z2) of M vanishes.

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For example, mathematicians ask whether or not a given oriented Riemannian manifold (M,g) admits spinors but can we start from a Pseudo-Riemannian manifold to see if admit the so-called pseudospinors?