As the title says, what methods exists for proving that a symmetric polynomial (or function) is Schur positive, perhaps involving extra parameters, in which case coefficients should be polynomials in the parameters with non-negative coefficients. This is what I have seen in the literature:

Representation-theoretical proof. Construct a (graded) $S_n$-module, and show that the Frobenius map of the decomposition into irreducibles gives the polynomial. This often involves finding recursions, and does not give formulas for the coefficients.

*Examples:*Modified Macdonald polynomials, and things in diagonal harmonics. This MO-post.RSK-type proof. This essentially gives a way to convert words to semi-standard tableaux, and is therefore a type of bijective proof.

*Examples:*skew-Schur functions (and thus the Littlewoord-Richardson rule).Via Gessel fundamental quasisymmetric expansion. It is usually easy to find the Gessel expansion of a combinatorially defined polynomial. Once this is known, the goal is to gather these pieces into Schur-polynomials. The 'Schur-expansion' is technically known from the Gessel expansion, but it is given in the form of functions $S_{\alpha}$, where one needs to modify the compositions $\alpha$ according to the 'slinky' rule to obtain partitions. This can introduce signs, that needs to be taken care of via a sign-reversing involution or similar.

Type-A crystal proof. If the polynomial is given as a sum over combinatorial objects, one can try to define a certain graph structure on these objects, fulfilling some combinatorial (fairly local) axioms (Stembridge did this characterization, if I recall). Each connected component of this graph will each correspond to a Schur polynomial in the Schur expansion. This is related to my old question, before I knew about crystals, and it turns out it is enough to consider three variables at a time (in the Stembridge axioms). Basically, if you can give a crystal graph in three variables, it should generalize to $n$ variables without any issue. The crystal structure is closely related to RSK, and also provides a representation-theoretical connection, as well as a (crystal) bijection to SSYTs.

*Examples:*Stanley symmetric functions. Dual Grothendieck.Dual equivalence graph, introduced by S. Assaf. Similar idea as crystals/RSK/Gessel. From the fundamental quasisymmetric expansion, define a graph structure that gathers these pieces into Schur-positive parts. Example: This article, which has a non-symmetric counterpart as well.