Robert Bryant's answer here ( https://mathoverflow.net/a/149496/85500 ) states that any orientable 3-manifold is parallelizable.
Previously I was under the impression that only closed (compact & without boundary) orientable 3-manifolds were necessarily parallelizable.
I have also heard somewhere (unfortunately I don't remember the source) that most noncompact 3-manifolds are parallelizable, but even if this statement is correct in some sense, I have no idea whether this "most" was meant informally or if this statement can be interpreted in a technical sense.
In either case, my knowledge of differential/algebraic topology is nowhere near my knowledge of local differential geometry so I do not know any proofs and would probably have difficulty understanding one anyways.
Primary motivation for asking this question is to try to understand how restrictive is demanding spacetimes in general relativity to be parallelizable. If we want to have a well-posed initial value problem, spacetime must be of the form $\mathbb R\times\Sigma$, where $\Sigma$ is a 3-manifold, so it is parallelizable iff $\Sigma$ is parallelizable.
It is also known that a spacetime admits spin structures if and only if it is parallelizable. So if one wishes to include fermions on spacetime, one wants a parallelizable spacetime.
Question: So what is the proper statement about the parallelizability of orientable 3-manifolds, especially in regards to non-compact ones?