The maximum number of vertical independent vector fields on the tangent bundle Let $M$ be a differentiable manifold.
Is there a name 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 name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$ and whose mutual flows commute, i.e. they are vertical and have pairwise zero Lie bracket? What kind of characteristic classes can be used to compute such quantities? What are these maximum numbers for $M = S^n$?
Edit:(After the answer by Michael Albanese)
The vertical rank of  $TM$ is the maximum number of  independent commuting vertical vector fields on $TM$. the rank of $M$ is the maximum number of  independent commuting vector fields on $M$?This terminology coined by Milnor

Question: Is the vertical rank of $TM$  equal to the rank of $M$?

 A: I will address the first version of your question (i.e. no conditions on commuting flows).
A vector bundle $E \to B$ admits $k$ linearly independent vector fields if and only if $E$ has a subbundle isomorphic to $\varepsilon^k$, the trivial rank $k$ bundle. The largest such $k$ is called the span of $E$. If $E$ has rank $n$ with span $k$, then $w_i(E) = 0$ for $i > n - k$ and $p_i(E) = 0$ for $i > \lfloor\frac{1}{2}(n-k)\rfloor$.
If $\pi : TM \to M$ denotes the natural projection, then the subbundle of $TTM$ consisting of vectors tangent to the fibers of $\pi$ is precisely $\ker(d\pi)$. Your first question can be rephrased as: what is the span of $\ker(d\pi)$? Note that $\ker(d\pi) \cong \pi^*TM$, and hence $\operatorname{span}(\ker(d\pi)) = \operatorname{span}(\pi^*TM)$. In general, $\operatorname{span}(f^*E) \geq \operatorname{span}(E)$ but in our case, $\pi : TM \to M$ is a homotopy equivalence, so we obtain $\operatorname{span}(\pi^*TM) = \operatorname{span}(TM)$ which we often call the span of $M$.
The span of a manifold is very difficult to calculate in general. For spheres, the problem was resolved by Adams in 1962 - note, the characteristic class conditions mentioned above tell us nothing here as spheres are stably parallelisable. The span of $S^{n-1}$ is $\rho(n) - 1$ where $\rho(n)$ denotes the $n^{\text{th}}$ Radon-Hurwitz number, defined as follows: if $n = 2^{4a + b}c$ where $a, b, c$ are non-negative integers, $0 \leq b \leq 3$ and $c$ is odd, then $\rho(n) = 8a + 2^b$.
