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](https://www.ams.org/journals/tran/1987-304-01/S0002-9947-1987-0906820-7/home.html). 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.