The accurate identification of the local configurations of defects in random alloys poses a challenge to experimental characterization methods. In this thesis, first principles computational approaches are implemented to solve this problem. In particular, three methods -- special quasirandom structures, real space Green's functions, and generalized force constants -- are combined as a means to deduce the phonon density of states (DOS) in a quasi-random In0.5Ga0.5As ternary alloy. This alloy has generated a lot of industrial interest as a potential replacement for silicon MOSFETs in low power, low voltage devices due to their high carrier mobility and drift velocity. We focus on the effect on the phonon DOS of including a number of commercially relevant defects and common dopants. We calculated all possible local configurations, together with an estimation of the respective strain fields for common dopants (Si, Se, and Te) and a cation vacancy defect within a quasi-random arrangement of an In0.5Ga0.5As alloy. The randomness of the cations in the alloy is important since this is believed to be the experimentally relevant configuration of the alloy, but one which is difficult to simulate computationally. Local vibrational modes of Si dopants occur at frequencies ranging from 22.88 THz to 29.05 THz, depending on the specific local environment. Se dopants have more intense local vibrations, occurring at lower frequencies ranging from 18.66 THz to 23.86 THz. Neighboring atoms of the vacancy respond much more weakly than those of Si and Se to the introduction of the defect to the vibrational density of states, where more than half of them show stronger global oscillations instead of local vibrations. Our results can serve as a guide for future experimental investigations, and our method can be generally applied to other semiconductor alloys.