It is well known that, by a theorem of Grothendieck, every vector bundle (always assumed coherent in this question; and everything is over the complex numbers) splits as a direct sum of line bundles.

This is not true in families, though, not even locally on the base of the family, as it is possible to construct flat $1$-parameter families of VB that have generically the same splitting type and "jump" to a different splitting type at the special fiber. This prevents the moduli stack $\mathfrak{M}$ of *all* VB on $\mathbb{P}^1$ (or the corresponding functor $F$ of isomorphism classes) to be separated (in the sense of valuative criteria). Here we took the notion of isomorphism of families to mean, as usual: the families $\mathscr{E}$ and $\mathscr{E}'$ of VB on $X$ parametrized by $T$ are isomorphic if there is a line bundle $L$ on $T$ such that $\mathscr{E}\simeq\mathscr{E}'\otimes L$.

For a smooth projective curve $X$, the moduli functor $F^s$ of *stable* bundles is representable by a smooth separated scheme $M^s$, and the stack $\mathfrak{M}^s$ of stable bundles is a gerbe over the latter. The moduli functor $F^{ss}$ of isomorphism classes of *semistable* bundles has a non separated coarse moduli space with proper connected components, and collapsing $S$-equivalence classes in the sense of Seshadri fixes this: now the functor $F'^{ss}$ of such $S$-equivalence classes of semistable bundles has a coarse moduli space (each connected component of) which is a projective variety (in particular separated).
By Grothendieck's theorem, the relation of S-equivalence on $\mathbb{P}^1$ coincides with the relation of isomorphism, so we can avoid talking of $S$-equivalence in this case. Also, there are no stable VB on $\mathbb{P}^1$ (of rank $>1$).

In what follows the curve $X$ is $\mathbb{P}^1$.

Since every VB is uniquely determined by its splitting type, and there is a countable set of choices for the possible line bundles occurring as summands, each connected component of $M^{ss}$, which is a variety, has to be a reduced point $\mathrm{Spec}(\mathbb{C})$.

Recall we let $\mathfrak{M}$ be the stack in groupoids of *all* VB on $\mathbb{P}^1$, with isomorphisms of families as $1$-arrows, and $F$ the corresponding functor of isomorphism classes.

Q.1:How does $\mathfrak{M}$ look like?

Q.2:Does $F$ have a coarse moduli space $M$? If yes, how does it look like?

It seems possible to me that an answer to question 1 or 2 has something to do with the partial order structure on the set $\boldsymbol{\mathrm{HNP}}$ of "Harder-Narasimhan polygons" (see Shatz, *The decomposition and specialization of algebraic families of vector bundles*). Maybe the specialization order on (field-valued) points of $M$ (or of $F$, or of $\mathfrak{M}$) reflects somehow the structure of the lattice $\boldsymbol{\mathrm{HNP}}$?