  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 $\beta$-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 $\beta$-score tests outperform any classical test.
