This project report entitled by 00Spectral analysis of singularities of distributions00 explored how singularities appeared in the theory of distribution. The main objective of the study is to detect singularities of distributions from both smoothness and analytic point of view. And those singularities that we are talking about can be detected by using the conic set which tells us the direction where the singularities are coming from and using the singular support of a given function which describes the location of singularities. In the study we used a new set called the wave front set which explored both the location and direction of singularities.