One way to rigorously define spinor fields on metric manifolds is through the language of associated bundles. Namely, we have a principal bundle $P \overset{\pi}{\longrightarrow} M$ over $\mathrm{Spin}(n)$ (or whatever other signature) that is a $\rho$-equivariant lift of the orthonormal frame bundle $OFM \rightarrow M$ through the map $\phi: P \rightarrow OFM$, where $\rho: \mathrm{Spin}(n) \rightarrow SO(n)$ is the double-cover map. $(P,\phi)$ prescribes the spin structure.
The associated bundle is then constructed with a typical fiber $F$, which is a representation space of $\mathrm{Spin}(n)$, and where the field takes its values. The transformation behavior of the objects is at that point fixed through the left and right actions of the group on these bundles. The fields are then taken as sections of the associated bundle, which can equivalently be expressed as $\mathrm{Spin}(n)$-equivariant functions on the principal bundle.
The part that is unclear to me is how these are related to the usual spinors that we see in physics. In particular:
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The general construction doesn't specify the choice of spin structure $(P, \phi)$. Are there any concrete examples for this choice?
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To get 'fields' (in the physics sense) defined on $M$ one usually needs a local trivialization of the principal bundle. How would this be provided in practice?
For tangent vectors, for example, the principal bundle is the frame bundle $FM$, and one can induce the trivialization by a chart $(U, x)$ on the base manifold $M$, namely by giving the section $\alpha: U \rightarrow FM,\,\; p \mapsto (\partial / \partial x)_p$, but this doesn't even port over to $OFM$ since $(\partial / \partial x)_p$ are not necessarily orthonormal.
Best Answer
A spin structure is required to allow for a principal bundle with gauge structure group a spin group. This is already assumed in the general construction you've outlined. Moreover, I expect that this condition is fulfilled over a flat space which is why it isn't usually mentioned. It's only when you begin to do QFT over curved space that this subtlety begins to be important.
The bundles in physics are what mathematicians generally call fibre bundles. This means they are exactly bundles that are locally trivialisable. This includes principal bundles. You do not need to specify a local trivialisation to obtain a field. They are what mathematicians call sections and these are defined without asking for a local trivialisation. You do need them, however, to work locally. That is to obtain a bundle chart. Bundle charts are equivalently, local trivialisations.
There isn't a canonical way of choosing a section in a fibre bundle. However, physicists often talk about "choosing a gauge" and this is physics-speak means picking out certain sections that simply the equations of motion. This os amply explored in thr physics literature. See for example, Lorenz or Coulomb gauge.