State-of-the-art accelerators at energy and intensity frontiers require increasingly bright and powerful particle beams. In conventional linear lattices, intense beams suffer from collective instabilities, resulting in beam losses and maximum beam intensity limits. This thesis presents experimental studies of a novel lattice design concept, the nonlinear integrable optics (NIO), aimed at enhancing beam stability limits with little to no beam dynamics degradation. Single-particle beam dynamics measurements of two NIO devices, the quasi-integrable octupole system and the fully integrable Danilov-Nagaitsev system, were carried out at the purpose-built Integrable Optics Test Accelerator (IOTA) at Fermilab. Their simulation, hardware design, and commissioning process are presented. Extensive model and analysis algorithm development and benchmarking is described. Electron beam data from two scientific runs is analyzed, yielding frequency and phase space dynamics consistent with models. These results demonstrate viability and advantages of the NIO design, providing the groundwork for proton studies in the strong space-charge regime and future integrable accelerators.