A general rule of thumb for any correlation:

Pearson

• parametric
• 'large' (representative) sample size
• data follows a normal distribution
• data is typically continuous

On the last point, if one conducts a linear regression where the predictor is a factor, it's the exact same as performing a Pearson correlation where that same factor is encoded numerically - see A: In GSEA, how can one "correlate" gene expression with categorical phenotype data

In an RNA-seq context, 'large' can simply mean a study comprising 5 biological replicates (some would say 3).

Kendall | Spearman rank correlation

• non-parametric
• small or large (i.e. any) sample size
• skewed data / data does not follow normal distrubtion
• ordinal or continuous data

Kevin