One application that I have in mind is in countable model theory:
Vaught's conjecture: If $T$ is a complete countable first order theory, then the number of nonisomorphic countable models of $T$ is either countable or $2^{\aleph_0}$
Shelah and Harrington proved the conjecture for $\omega$-stable theories, but the general problem is still open (it may be the longest standing open problem in model theory, but I'm not really sure).
However, it turns out that a more general problem can be stated in terms of Polish group actions:
Topological Vaught's conjecture: Let $G$ be a Polish group with a Borel-measurable action on a Polish space $X$ and let $A$ be a Borel set of orbits, then either $A$ contains at most countably many orbits, or $A$ contains a perfect set $B$ such that each two elements of $B$ are in different orbits.
Since isomorphism classes of $L_{\omega_1,\omega}$ theories correspond to certain orbits of Polish group actions, TVC implies VC.
See this paper for more background: http://www.jstor.org/stable/2275907