Mozilla(Spider Monkey JS engine) uses mergeSort. You can see the code written for it in C in the Mozilla repository: https://dxr.mozilla.org/seamonkey/source/js/src/jsarray.c
WebKit(Chrome, Safari, etc) do not directly use a sorting algorithm, instead they choose the algorithm based on element types and length of the array. For example,
Numeric arrays use C++ Std library's quick sort function.
Non-numeric arrays use merge sort.
In some other cases it uses selection sort.
It depends on the datatype and size of array elements which algorithm will be used to sort the array.