We explore the dynamics of integrable and chaotic quantum many-body systems with a focus on universal structures known as caustics, which are a type of singularity categorized by catastrophe theory. Papers I and II study light cones in quantum spin chains, which we show are caustics and therefore inherits specific functional forms. For integrable systems, the edge of the cone is a fold catastrophe, making the wavefunction locally of Airy form. We also identify the cusp catastrophe in the XY model, thus the secondary light cone is a Pearcey function. Vortex pairs appear in the dynamics, are sensitive to phase transitions, and permit the extraction of critical scaling exponents. In paper II we use a Gaussian wavefront form to distinguish integrable and chaotic models. Writing the wavefront as exp[−m(x)(x − vt)2 + b(x)t], the scaling of coefficients m(x) and b(x) is the diagnostic. The local Airy function description in free models leads to a power-law ∼ x^{−n/3} scaling, while for the chaotic case the scaling is exponential ∼ e^{−cx}. In Paper III, we study the function Fn(t) = <(A(t)B)^n>, a generalization of the four-point out-of-time ordered correlator (OTOC) F2(t), for an integrable system and show that the function Fn(t) can be recast as the return amplitude of an effective time dependent chaotic system, exhibiting signals of chaos such as a positive Lyapunov exponent, spectral statistics consistent with random matrix theory, and relaxation. In Paper IV we perform a comprehensive investigation of caustics in many-body systems in (1+1)- and (2+1)-dimensional Fock space and time. We show how a hierarchy of caustics appear in the dynamics of many-body models, using two- and three-mode Bose-Hubbard models as guiding systems. We show that, in the case of the trimer, high dimensional caustics appear and are organized by the catastrophe X9. / Thesis / Doctor of Philosophy (PhD)