Let $M$ be a smooth manifold of dimension $n.$ It is well known that it is not allways possible to find such global vector fields $X_1,\ldots, X_n$ which would be a basis for the tangent space at each point of $M.$ Indeed, that can only happen for parallelizable manifolds $M.$
But how about if we allow a larger number of fields, so that despite a few of them having to be linearly dependent at any given point $p\in M,$ they will still span $T_pM$? I feel like such a set of vector fields should always exist, even under the added requirement that it should be finite, but I don't know how to prove either ...
Let $f:M\to\mathbb R^N$ be an embedding (such a thing exists for $N$ sufficiently large). Let $X_i$ be the normal projection of the constant "coordinate" field $e_i$ on $\mathbb R^N$ to the tangent subspaces to $M$. This gives (by pullback) $N$ vector fields on $M$ which span the tangent subspaces at each point.
For example, $S^2$ embeds in $\mathbb R^3$ and this gives three tangent vector fields on $S^2$ which span the tangent space at each point. You should write them explicitly, for fun.