I believe I misunderstood your question the first time around.
As pointed out by others, there is usually not a recipe for how or when to perform multiple comparison testing. In order to have a better understanding for what might be appropriate in your case, we would need more information about your study design and what constitutes a "test."
Suppose you have a reference sample, Sample0, which has a list of numbers associated with it. In a series of tests, you compare SampleA, SampleB, and SampleC to Sample0.
Because each test is different, it has different statistical assumptions, statistical power, different robustsness to deviations from those assumptions, and perhaps even a different statistical test for calculating a p-value. E.g. one might be a t-test, another could be an F-test, another a Chi-Squared test, etc.
In such a case, the tests do not form a family, but the samples do. Since the samples are drawn from hypothetical parent populations that have something in common, there is always a random chance that non-significant differences will test as significant, or visa-versa. I would personally apply something like the BH correction across samples.
Meanwhile, suppose the samples were a completely heterogeneous collection, say individual colonies of different animal species, but the tests were repeated measurements of heart rate over the course of a year or something. The p-value compares the colony heart rate between males and females. In this case, column-wise multiple comparisons might make more sense. We want to control for false positives in the tests within each colony. But the colonies themselves have completely different statistical makeups.
I hope that helps.