The entropy of a multivariate Gaussian is given at https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Entropy
as $$\frac{1}{2}\ln((2\pi e)^n |\Sigma|).$$
Here $n$ is the dimension of the random vector.
What happens when the covariance matrix $\Sigma$ is singular? Imagine, for example that two rows of the matrix are identical. Intuitively I feel this shouldn't reduce the entropy by too much if $n$ is large but the formula no longer applies.