How to assess normality under the OLS assumptions? When we have a multivariate regression function, which assumption has to hold so that the OLS assumptions are not violated:
a) each variable must follow a normal distribution
b) all variables together must follow a normal distribution
c) only the dependent variable must follow a normal distribution
d) the residuals of the regression function must follow a normal distribution?

I'm a little confused, to be honest. Thanks in advance for your answers!

The first two are totally wrong but are common misconceptions about the normality assumption in OLS regression (when we choose to make such an assumption, which we don’t have to do).
There is no distribution assumption about the predictor variables, and there certainly is no normality assumption. For instance, ANOVA can be seen as an OLS regression, and ANOVA uses binary predictor variables, which certainly aren’t normal. Thus, A is false.
Since none of the features have to be normal, the features do not have to be jointly normal, and B is false.
C represents another common misconception. The marginal distribution of $y$ has no particular assumption. It is common to see people transform a non-normal $y$ to achieve marginal normality. While there are legitimate reasons for transforming $y$, it is far from a necessity.
D is the closest to a correct answer, but I dispute it on a technicality. The observed residuals are not normal. It is the unobserved error that is normal. The residuals are a discrete set that cannot be normal. However, many people use “residuals” and “errors” as synonyms. This slang tends not to cause problems in practice, but that’s only after we know the real definitions.
When the errors are independent and identical Gaussians, the OLS solution corresponds to maximum likelihood estimation of the regression parameters, and that allows us to do inference on the coefficients and on nested models in the usual way with t-stats and F-stats, respectively.
(The t-stats and F-stats wind up being pretty robust to violations of the normality assumption, particularly with large sample sizes.)
