In this article, we introduce two new classes of single-valued contraction mappings within the framework of S-metric spaces. For the first class, called S-polynomial type contraction mappings, we establish two fixed-point theorems that demonstrate the existence and uniqueness of fixed points. We begin by considering the case where the mapping is continuous and then relax this continuity requirement. Importantly, we also derive Banach’s fixed-point theorem as a special case within the context of S-metric spaces. The second class, termed almost S-polynomial type contraction mappings, builds on the concept of almost contraction mappings introduced by Berinde. We establish two fixed-point theorems that generalize Berinde’s results in S-metric spaces. Additionally, we introduce a condition for uniqueness and prove a theorem for a unique fixed point. Several examples are provided to illustrate the significance of our generalizations. Finally, we demonstrate an application of the main result by describing the solution of the integral equation.