If you have multivariate data than you will want to use a PERMANOVA probably. For instance, if you have community matrices from different sampling sites and you want to know if the two communities differ.

If you have a single-response variable then you could use a KW test. Note that if you have a single-response variable, you might not really need to do a KW test unless your data are really wonky. ANOVA is robust to modest violations of assumptions.

If you do a PERMANOVA then it is wise to test if the sampling groups being compared have similar dispersion. You can use the betadisper function in the vegan package to do this in R. Hope this helps. Good luck.

Without knowing your data and your research aims, this is not an easy question to answer.

Have you considered applying a generalized linear model (GLM)? The advantage of this approach is, that it allows you to fit response variables that have error distributions other than a normal distribution while still being parametric.

All the best!

My Dear

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See

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