Shelah's black box is used widely in solving algebra problems. One example that I like is the following work of
Dugas and Göbel
All infinite groups are Galois groups over any field. Trans. Amer. Math. Soc. 304 (1987), no. 1, 355–384.
In this paper, Shelah's black box is used to prove the infinite analogue of the still unsolved Hilbert-Noether inverse Galois problem.
Also a nice and short reference is The uses of set theory by Roitman. The following is taken from Mathscinet:
The author's purpose is to show how modern set theory is relevant to other parts of mathematics, particularly areas not ordinarily regarded as close to set theory (in contrast to, e.g., general topology). After a brief section on set-theoretic background, most of the paper consists of specific examples of connections between set theory and other areas. Two of the examples are mentioned only very briefly, because thorough expositions of them exist elsewhere. These are the independence of Kaplansky's conjecture on automatic continuity of certain Banach space homomorphisms and Whitehead's conjecture about freeness of abelian groups. The other six examples are presented in somewhat more detail, including the basic ideas behind the proofs, in approximately one-half to one (large) page per example. The first example concerns the work of G. Weiss, S. Shelah, and the reviewer, relating properties of ideals of compact operators on Hilbert space to the combinatorial principle of near coherence of filters and establishing the consistency and independence of this principle. The second is a characterization by J. Steprāns of free abelian groups as those admitting discrete norms. The third is Shelah's theorem that the fundamental group of a nice space is either finitely generated or of the cardinality of the continuum. The fourth is an independence result arising in strong homology theory, where a result proved by S. Mardešić and A.V. Prasolov under the continuum hypothesis was shown by A. Dow, P. Simon, and J. Vaughan to be unprovable in ZFC. The fifth is an example, due to Shelah and Steprāns, of a nonseparable Banach space where every linear operator is a scalar multiplication plus an operator with separable range. The last concerns R. Laver's work, arising from large cardinal theory, on the free left-distributive algebra on one generator. The paper includes references to either the original sources or surveys for each of the examples presented. (P. Dehornoy has recently shown that the irreflexivity of the ordering in the author's last example can be proved without large cardinal hypotheses; yet another connection with mainstream mathematics is exhibited by the title of Dehornoy's as yet unpublished paper: "Braid groups and left distributive structures''.)