This dissertation investigates the application of concepts and tools from descriptive set theory to address classification problems in various areas of mathematics, specifically focusing on Polish modules and the complexity of equivalence relations arising in model theory. First, we address the structure and classification of Polish modules. In joint work with Sławomir Solecki, we introduce a general method for constructing Polish modules over arbitrary subrings of $\mathbb{Q}$ using ideals of subsets of $\mathbb{N}$ and sequences in $\mathbb{N}$. We utilize this method to resolve a question posed by Frisch and Shinko, demonstrating the existence of uncountably many Polish $\mathbb{Q}$-vector spaces that embed into $\mathbb{R}$ yet are mutually incomparable with respect to continuous $\mathbb{Q}$-linear embeddings. Second, we refine the analysis of equivalence relations in model theory. In collaboration with Sławomir Solecki and Bilge Köksal, we propose a general framework termed "zigzag" (denoted by $\curlywedge$) for comparing the complexities of $F_\sigma$ equivalence relations on zero-dimensional compact spaces. This framework generalizes the notion of homeomorphism between quotient spaces and provides a finer distinction than standard Borel bireducibility. We apply this to the study of Lascar strong types, constructing examples of theories where some Lascar strong types belong to the same Borel bireducibility class but are distinct under the zigzag relation. Finally, we examine classes of sets, identified with their singletons, up to bisimulation. We establish a canonical definition of a relation $\in'$ on these classes and demonstrate that they satisfy the set-theoretic axioms of extensionality, pairing, and union with respect to $\in'$.