Nowadays copulas are extensively used in economic and finance applications, with the Gaussian copula being very popular despite ruling out non-linear dependence, particularly in the lower tail. We derive computationally simple and intuitive expressions for score tests of Gaussian copulas against Generalised Hyperbolic alternatives, which include the symmetric and asymmetric Student t, and Hermite polynomial expansions. We decompose our tests into third and fourth moment analogues, and obtain more powerful one-sided Kuhn-Tucker versions that are equivalent to the Likelihood Ratio test, whose asymptotic distribution we provide. We conduct detailed Monte Carlo exercises to study our proposed tests in finite samples. (Coauthor: Enrique Sentana.)