The field of quantum simulations has achieved a remarkable success through thedevelopment of highly controllable and accessible quantum platforms, which pro-vide insights into the microscopic properties of complex large-scale systems thatare otherwise difficult to analyze. Many of the platforms utilized in this pursuit arederived from the field of atomic, molecular, and optical physics. One particularlypopular candidate is provided by trapped ions, whose vibrational and electronicdegrees of freedom can be effectively combined through laser pulses to engineerdesired model Hamiltonians or quantum circuits. Trapped ions constitute as wellthe basis for modern atomic clocks, the most precise frequency standards currentlyavailable. They find further applications in metrology, geodesy, and fundamentalphysics experiments.In this Thesis, we investigate the dynamics of vibrational modes in trappedion crystals, utilizing them as a versatile platform to explore various many-bodyphenomena.We first focus on the expansion dynamics of local excitations and on heattransport within ion crystals hosting structural defects that undergo a sliding-to-pinned transition. We observe a significant reduction in conductivity whenthe crystal symmetry is spontaneously broken during the transition, and showthat resonances between crystal eigenmodes lead to distinct softening signaturesassociated with energy localization. We then delve into the effects of thermal andquantum fluctuations on the vibrational modes of ion crystals near two distinctstructural transitions. We observe the emergence of a prolonged symmetric phasestabilized by thermal and quantum fluctuations, and develop effective theories thatreduce the degrees of freedom to the modes that drive the transitions.Finally, we discuss how to engineer spin-orbit coupling and on-site interactionenergies for vibrational quantum excitations using two different external drivingschemes. While the simulation of spin models with ions typically involves the useof two electronic states, we propose interpreting the two local oscillation modesin an ion crystal as a pseudospin. We show how using Floquet engineering ideasallows for spin flips in Coulomb-induced vibron hopping, resulting in a non-trivialcoupling between spatial motion and spin evolution, that results in a markedly non-Abelian dynamics. Subsequently, we explore the simulation of Hubbard models intrapped ions by coupling the vibrational Fock states to an internal level system.Our findings include the observation of bound states in the strong interaction limitof the resulting Jaynes-Cummings-Hubbard model.By investigating these topics, we aim to contribute to the understanding ofvibrational dynamics in trapped ion crystals, and shed light on their potential forsimulating condensed matter systems, offering insights into phenomena that areotherwise challenging to explore.