This work provides calculus rules for the Fréchet and Mordukhovich subdifferentials of the pointwise supremum given by an arbitrary family of lower semicontinuous functions. We start our study by showing fuzzy results about the Fréchet subdifferential of the supremum function. Subsequently, we study the Mordukhovich subdifferential of the supremum function in finite- and infinite-dimensional settings. Finally, we apply our results to the study of the convex subdifferential; here we recover general formulae for the subdifferential of an arbitrary family of convex functions.