Topology and superconductivity are quintessential quantum orders in condensed matter physics. Their interaction yields the exotic state of topological superconductivity, predicted to host Majorana modes useful for fault-tolerant topological quantum computations. However, most experimental claims of topological superconductivity have measured the coexistence of the ingredients rather than the products of a topological superconductor; smoking-gun signatures of Majorana modes have faced skepticism; and trivial explanations have been plausibly proposed. Broader study of candidate topological superconductors is therefore needed to validate the physics of these systems, and new experimental probes must be developed to provide new perspectives on the challenges facing this field. In this work, we utilize linear and nonlinear terahertz spectroscopy techniques to probe candidate topological superconductor systems. We detail four distinct studies that can be categorized by a Punnet square of the topological superconductivity platforms (proximity effect and connate) and experimental method (linear and nonlinear terahertz). For the linear-proximity pairing, we study the SmB$_6$ (candidate topological Kondo insulator) / YB$_6$ (BCS superconductor) heterostructure, demonstrating the presence of an interface surface state predicted to host topological superconductivity. For the linear-intrinsic pairing, we study the widely-accepted connate topological superconductor Fe(Te,Se), finding evidence that the superconducting state is disordered. For the nonlinear-proximity pairing, we study the topological insulator Bi$_2$Se$_3$, observing intrinsic terahertz second harmonic generation for the first time, originating from the topological surface state. For the nonlinear-connate pairing, we again study Fe(Te,Se), discovering a novel terahertz third harmonic response that appears to be poorly described by the conventional Higgs mode oscillation. Ultimately, this work increases the understanding of various topological superconducting platforms as realized in specific material systems, establishes linear and nonlinear terahertz methods as effective probes of the interaction of topology and superconductivity, and paves the way toward more intensive investigations of topological superconductivity by these methods. While this work does not propose overarching conclusions concerning topological superconductivity, each study sheds new light upon the interaction of topology and superconductivity in these material systems and validates linear and nonlinear terahertz spectroscopy methods in their study.