JavaScript uses the TimSort method for Sorting Arrays in Chromium V8 engines, available since V8 v7.0 & Chrome v70, as of Sept' 2018.
Numeric arrays or arrays of primitive type were used to be sorted using the C++ standard library function std::qsort which used to implement some variation of Quicksort (usually Introsort), before Chromium V8 shifted to TimSort.
Contiguous arrays of non-numeric types were stringified and sorted using mergesort, if available (to obtain a stable sorting) or qsort if no merge sort is available.
For other types (non-contiguous arrays) and presumably associative arrays, WebKit used either selection sort, which they call minsort or, in some cases, via AVL trees.