Mechanisms generating free subgroups of Artin braid groups Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that sufficiently high-powers of hyperbolic elements in a Gromov-hyperbolic group generate a free group arises via ping-pong. I know of no tree to represent the situation. 
To the experts, the following question is surely either an obvious yes or no: 
Can we explicitly describe all mechanisms by which finitely generated free subgroups of the Artin braid groups $B_n$ (for $n=2,3,\ldots$) arise ? 
More specifically, seeing $B_n$ as the isotopy space of all $n$-pointed braids in the closed 2-disk, can we characterize all configurations generating the rank $k$ free group?
Much less specifically, do we know, say, that all finitely-generated free subgroups arise from ping-pong and can we describe all ping-pong games? 
 A: A theorem of Leininger-Margalit says that any two elements of the pure braid group either commute or generate a free group; see here.  But I don't think there is any hope for classifying free subgroups that are generated by more than $2$ elements.
A: This is a very extended comment to your question. 
Your question does not have "obvious yes or no" answer; in part, this is because you asked 3 somewhat different questions: 


*

*"Can one explicitly describe all finitely generated free subgroups..." In this form, the question has no answer since the word "explicit" is meaningless in this context. 

*Your second question "can we explicitly enumerate/characterize all configurations which generate the rank k free group" is a bit better, since one can interpret it as a request for an algorithm that, given as its inputs tuples $(w_1,...,w_k)$ of words in the standard generators of $B_n$, would determine if they generate rank $k$ free subgroups of $B_n$. Existence of such an algorithm is unknown for $n\ge 6$. The reason is that $B_n$ contains $F_2\times F_2$ and, hence, also contains Mikhailova subgroups. In particular, it is undecidable if a given set of $k$ elements in $F_2\times F_2$ generates $F_2\times F_2$ or not.     

*Lastly, you are asking something about ping-pong. What this "something" is, is very much unclear. Every free group $F_k$ has a ping-pong description (this is almost a tautology). On the other hand, the "collection" of such ping-pongs is not even a set! One can ask, however, a meaningful question restricting to "Schottky ping-pongs", by considering the action of $B_n$ on a certain topological sphere $S=S^{n-3}$, the space of projective classes of measured laminations on $n+1$ times punctured sphere. A Schottky ping-pong on $S$ is given by a collection of disjoint compact subsets $C_1, C_1',...,C_k, C_k'$ and elements $g_i\in B_n$, so that each $g_i$ sends the interior of $C_i$ homeomorphically to the exterior of $C_i'$. Then the subgroup $<g_1,...,g_k>\subset B_n$ is free of rank $k$ and is necessarily necessarily purely pseudo-Anosov; such subgroups are Schottky subgroups in $B_n$ by analogy with Schottky groups acting on hyperbolic spaces (and their ideal boundaries). It is also unknown if $B_n$ (or any mapping class group for this matter) contains a purely pseudo-Anosov free subgroup which is not Schottky.  Note that already $PSL(2,C)$ contains free subgroups of rank 2 which are purely hyperbolic and not Schottky. 
A: A couple of remarks: 
If a subgroup is torsion-free, and intersects the pure braid group $P_n$ in a free group, then the group is also free. So the question can be interpreted in $P_n$. 
There is the Birman exact sequence $F_{n-1} \to P_n \to P_{n-1}$, which is obtained as deleting a puncture, and has kernel the fundamental group of an $n-1$-punctured plane. There is such a free subgroup for each puncture, and these subgroups are not ``quasiconvex" in any sense since they are normal. So I don't think there is some sort of ping-pong description of these from some action of $P_n$ on a nice space. 
On the other hand, if the image of the subgroup generated by $g_1,\ldots,g_k$ in $P_{n-1}$ is free of rank $k$, then so is the subgroup of $P_n$. So I could imagine some inductive description of the free subgroups of $P_n$. 
Suppose the image in $P_{n-1}$ is free. Then there is a Nielsen transformation of $g_1,\ldots,g_k$ such that $g_1,\ldots,g_j$ have image in $P_{n-1}$ generating a rank $j$ free subgroup, and $g_{j+1},\ldots,g_k$ have trivial image in $P_{n-1}$. Then the question is what is the extension of the free group $\langle g_1,\ldots,g_j\rangle$ by the normal subgroup of $\langle g_1,\ldots,g_n\rangle$ generated by $g_{j+1},\ldots, g_n$? I wouldn't expect a simple answer to this question though, so this is probably a difficult question. Also, the image in $P_{n-1}$ might not be free.   
