I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly:

What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin (or not spin) (and why)?

For comparison, I'd consider the cylinder and the mobius strip to be "good" examples of orientable (or not) bundles.

I've read the answers to Classical geometric interpretation of spinors which are helpful, but I'd like specific examples (non-examples) to think about.