Of course, this is a very broad topic. Nevertheless, is there a ready-made set of tests that shows the advantages and flaws of well-known sorting algorithms in the context where the std::sort or qsort() are used?

Actually I created some improvements to a quicksort-based algorithm and it work pretty for target scenarios (i.e. faster than std::sort, that based on introsort). So I want to check/estimate these improvements on a good suite of test datasets to get a better picture of the pros and cons.