From my perspective, either you have data and you have an idea what sort of distributions you're working with, or you don't, and you should fix that problem first rather than going down the theory rabbit hole.

With data in hand, a skew/kurtosis scatter plot is a good way to gauge the higher dimensional distribution of your data. Another option is to cluster the variables of the data set using something like HDBSCAN and color the plot points based on cluster membership.

If you have to go guessing distributions without evidence, you're better off choosing a low k student's T distribution (for robustness to outliers) or a gamma distribution (if you think your data might be skewed).