Even though the Rao's score tests are classical tests, such as the likelihood ratio tests, their application has been avoided until now in a multivariate framework, in particular high-dimensional setting. We consider they could play an important role for testing high-dimensional data, but currently the classical Rao's score tests for an arbitrary but fixed dimension remain being still not very well-known for tests on correlation matrices of multivariate normal distributions. In this paper, we illustrate how to create Rao's score tests, focussed on testing correlation matrices, showing their asymptotic distribution. Based on Basu et al. (2021), we do not only develop the classical Rao's score tests, but also their robust version, Rao's $β$-score tests. Despite of tedious calculations, their strength is the final simple expression, which is valid for any arbitrary but fixed dimension. In addition, we provide basic formulas for creating easily other tests, either for other variants of correlation tests or for location or variability parameters. We perform a simulation study with high-dimensional data and the results are compared to those of the likelihood ratio test with a variety of distributions, either pure and contaminated. The study shows that the classical Rao's score test for correlation matrices seems to work properly not only under multivariate normality but also under other multivariate distributions. Under perturbed distributions, the Rao's $β$-score tests outperform any classical test.