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.