Abstract: A vortex sheet is an asymptotic model of a parallel shear flow where the width of the vorticity layer is much smaller than the layer's characteristic length. Unfortunately, the equation governing the time evolution of vortex sheets in an inviscid incompressible flow is ill-posed in that smallest scales grow the fastest. Even starting with ananalytic initial profile, the sheet forms a curvature singularity in finite time. Theoretically, the sheet has a "weak" solution after the singularity time, but the precise nature of the solution is unknown. Since numerical methods fail to produce meaningful solutions, researchers turned to regularized vortex sheet motion. One common method has been to smooth the kernel in the Birkhoff Rott integral that determines the vortex sheet velocity. In this thesis, I study two classes of smoothed kernels numerically. Different algorithmic approaches are examined with the goal to reduce calculational costs. Numerical solutions are then obtained to suggest possible form of the limiting solution as the smoothing parameter tends to zero. Similarity studies is conducted to understand the connection between the smoothing parameter and time.