When correcting for multiple comparisons, do you correct all p-values or only significant ones? This feels like a really basic question, but I haven't found a definitive answer.
When correcting p-values for multiple comparisons, are you supposed to only run the analysis/correction on the p-values that are significant before correction, or do you run it on all of them?
Or does it depend on whether you are running a FWER (Bonferroni, Holm, etc.) or a FDR (Benjamini-Hochberg, etc.) correction?
 A: A general bonferroni correction requires that you correct based on the total number of tests that have been run. Given that corrections for multiple comparisons should be planned for in advance of the actual analysis, the results of the tests in question have no bearing on the validity of the multiple comparisons correction method.
Consider the case where you run 100 random tests, and you find that only one of them gives you significant results. If you only 'corrected' for that one test, then that one test remains significant even though you ran 100 tests, clearly this wouldn't make sense.
There are of course other ways to correct for random comparisons, especially if the tests are related in some form. However, all of the methods that I know of are independent of the actual result of the tests.
A: I have often interpretted a 0.06 $p$-value as "borderline" significant, especially in small sample size studies. If you are presenting multiple $p$-values for an analysis, I would strongly advocate presenting all $p$-values on the same scale. 
That means either corrected or not, but not a mixture thereof. This is misleading the reader and begs the question why one would even present $p$-values at all.
If you use a FDR approach, the raw $p$-values must be presented. In permutation testing or bonferroni you may either transform all the $p$-values onto a non-uniform scale with a family wise error rate of $\alpha$ or present the untransformed $p$-values and annotate which ones are actually significant at the $\alpha$ level, (so a 0.02 finding may not be significant in the family wise test).
A: Mensen's commment is incorrect. Whether controlling FWER or merely FDR, your corrections must account for ALL tests conducted. If you only input the p-values that are < .05, an FDR-controlling procedure will invariably find all of them significant at the .05 level (making FDR control pointless).
