If you want to guarantee the same order regardless of the order of elements in the input array, here is a stable sorting:
myArray.sort(function(a, b) {
/* Storing case insensitive comparison */
var comparison = a.toLowerCase().localeCompare(b.toLowerCase());
/* If strings are equal in case insensitive comparison */
if (comparison === 0) {
/* Return case sensitive comparison instead */
return a.localeCompare(b);
}
/* Otherwise return result */
return comparison;
});