YYYY-MM-DD for anything technical or for file file systems. MM/DD for everything else.
DD/MM has no advantage, the only reason to use it is familiarity. For single dates neither notation is more useful, however when dealing with multiple dates I'm far more likely to be interested in finding all the dates in a given month. Good luck scanning a list of dates for all those in for example July, with them written in DD/MM. MM/DD is also superior for date ranges. With 7/10 - 8/20 I can immediately see I'm looking at multiple months, 10.07 - 20.08 is far less obvious.
So really, either notation is only useful for conveying information to other people, and for that MM/DD is the only one that has actual advantages over DD/MM.