Tests for randomness are described, when the data consists of points on the circumference of a unit circle, and the alternative to the uniform distribution (randomness) is either a unimodal or a bimodal distribution: the von Mises distribution or an adaptation. Tables of significance points are given for the test statistics, and the power is used to give a table of sample sizes needed to detect a given degree of clustering, measured by the parameters of the distributions. This paper is a close parallel, for two dimensions, to Stephens [15], which gave corresponding tests and tables for three dimensions.