We use the collocation Trefftz method to solve many models of Laplace boundary value problems on polygons with different kinds of singularities.We study their convergent behaviors(exponential or polynomial)and their causes.It is uncovered that the relation between the order of convergence and the intensity of corner singularities.We also experience the transition behavior of the convergence from one order to another.Finally,we will apply our results to Trefftz method with enriched basis.