The present work deals with the local structure and the lattice dynamics in quasicrystals. The work consists of two parts. The first part shows the investigation of the local structure of i-MgZnY by EXAFS measurements at the Zn and Y absorption edges. Part two describes the results on the phonon density of states (VDOS) in i-MgZnY and d-AlNiFe. The experimental methods used in this work include inelastic nuclear resonant scattering of synchrotron radiation (INRS), inelastic neutron scattering (INS) and inelastic scattering of synchrotron radiation with nuclear resonant energy analysis (IãS. In part I, the structural investigations via the EXAFS at the Zn and Y absorption edges reveal strong similarities in the local Zn environment in the MgZnY system. In the literature a reversible transition is described between a hexagonal MgZnY phase (h-MgZnY) and the quasicrystalline i-MgZnY Phase. In contrast to the h-MgZnY phase there is in our model no Y in the first shell of the Zn atoms. These results are confirmed by a comparison of the EXAFS measurements at the Y absorption edge in h-MgZnY and i-MgZnY. In part II we present our studies of the phonon density of states (VDOS) in d-AlNiFe and i-MgZnY using inelasitic neutron scattering (INS) and inelastic ã-scattering. A comparison of the results for INS and IãS reveal both the existence of a so called boson peak, which can be described as an excess of the density of states compared to the Debye squared frequency law at low energies. Corresponding experimental results for the specific heat, the velocity of sound and phonon dispersion curves found in the literature show together with our experimental results the existence of localized vibrations in i-MgZnY giving rise to such boson peaks. During the analysis of the INS data in d-AlNiFe a new method for the determination of the partial atomic VDOS was used. The results for the partial 57Fe-VDOS could be confirmed by a independend INRS measurement on the same sample.