The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem E-F games usually give me a much better understanding of the structures I'm working with.
In principle, though, one could also use the Keisler-Shelah ultrapower theorem – "Two structures are elementarily equivalent iff they have isomorphic ultrapowers by some ultrafilter" (see Ultrafilters arising from Keisler-Shelah ultrapower characterisation of elementary equivalence; one direction is trivial, Keisler proved the other direction under GCH, and Shelah brought the proof down to ZFC). However, I know of no setting in which this is actually a better way to approach the specific problem of showing elementary equivalence. So, my question is:
If we want to show $\mathcal{A}\equiv\mathcal{B}$, is it ever efficient to go through the Shelah-Keisler ultrapower theorem?
I'm asking about Keisler-Shelah specifically, as opposed to some other result about elementary equivalence, because any K-S based approach to establishing elementary equivalence would presumably involve some set-theoretic combinatorics, and I'm generally interested in set theory cropping up in "concrete"(ish) questions. My suspicion is that the answer is "no," and that the work involved in showing that two structures have isomorphic ultrapowers would always subsume the work involved in establishing elementary equivalence, but I have no real evidence for this.
Best Answer
People in algebra use Keisler-Shelah a lot. For example, Malcev proved that for fields $F,K$ one has $GL_m(F)$ is elementary equivalent to $GL_n(K)$ iff $m=n$ and $F$ is elementarily equivalent to $K$. The idea is first you prove that this is equivalent to $M_m(F)$ is elementarily equivalent to $M_n(K)$. Then you take ultrapowers and use that taking matrices commutes with ultrapowers to deduce that $M_m(F)$ and $M_n(K)$ have isomorphic ultrapowers iff $m=n$ and $F,K$ have isomorphic ultrapowers, i.e. $F,K$ are elementarily equivalent.
There are other examples like this in algebra. A good reference is the paper On Malcev's theorem on elementary equivalence of linear groups by Beidar and Mikhalev (Contemp. Math. 131 Part I (1992), 29-35).
I believe Sohrabi and Miasnikov also use Keisler-Shelah in their work on elementary equivalence of nilpotent groups.