You can (and probably should) add a threshold to recursive algorithms like mergesort so that they don't end up doing recursive calls on very small arrays.

For arrays smaller than the threshold, insertion is faster than bubble sort. And if you really don't want recursion for large arrays to begin with, you have heapsort or Shell sort.

I don't think there is any practical case where using bubble sort makes sense, except "efficiency doesn't matter for my use case and this is the code I already have".

As in one of volumes of Knuth's The Art of Computing, Heap sort is in place and achieves the Gleason bound so can be regarded as not beaten by quick sort.