Let $M$ be a differentiable manifold.

Is there a terminology for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a terminology for the maximum number of globally defined independent vector fields on $TM$ whose mutual flow commute and they tangent to fibers of $TM\to M$, that is they are vertical and they mutually have zero lie bracket? What kind of characteristic classes can be used to compute such quantitties? What are these maximum numbers for $M=S^n$?