I'm reading Tu's Introduction to Manifolds section on oreintablility.
On an orientable manifold, if any vector field vanishes at a point $p$ (e.g. $S^2$), then the global frame $(X_1,X_2)$ that represents the orientation is necessarily discontinuous.
I think this is true because for an open set $U$ containing $p$, the orientation equivalence classes $[(X_{1,q},X_{2,q})]$ can't all be the same because $T_pM=\{0\}$, which has it's own convention $+$ or $-$ which is incompatible with $+[(X_{1,q},X_{2,q})]$ and $-[(X_{1,q},X_{2,q})]$
Is this correct, and is this conevention for orientation of $\{0\}$ the most common?
If you had a continuous global frame for the tangent bundle, then the vector fields that constitute the frame would be linearly independent at every point (because they form a basis for the tangent space at that point). In particular, none of them can vanish at any point.
This doesn't actually have much to do with tangent bundles. The same argument shows that, for an arbitrary vector bundle, if each continuous global section vanishes somewhere, then there cannot be a continuous global frame.