For rank tests (e.g. Mann-Whitney, Wilcoxon, Kruskal-Wallis):
There are also no formulas for sample size calculations. What to do? Streiner and Norman advise, “Determine the sample size from the equivalent parametric test and leave it at that. Recent work has shown that you don’t lose any power with these tests and when the data aren’t normal, they may even be more powerful than their parametric equivalent.” They provide two references for this conclusion: Blair and Higgins (1985), or Hunter and May (1993).