In recent years, the parameterized level set method (PLSM), which rests on radial basis functions in most early work, has gained growing attention in structural optimization. However, little work has been done to investigate the effect of the basis functions in the parameterized level set method. This paper investigates the basis functions of the parameterized level set method for topology optimization of continuums. The effects of different basis functions in the PLSM are examined by comparing the required storage, convergence speed, computational efficiency, and optimization results, with the benchmark minimum compliance problems subject to a volume constraint. Besides, several schemes to enhance computational efficiency are proposed. The study on examples with unstructured 2D and 3D meshes can also be considered as a tentative investigation of future possible commercial applications of this method.