Easiest way to determine the apostrophe position is to reorder the sentence or phrase so that the word requiring an apostrophe comes at the end. You'll find that the correct apostrophe position will be at the end of the final word (adding an 's' where it doesn't already have one), as follows:
The Putting Contest of the Men[***apostrophe will go here***] = Men's Putting Contest
The Putting Contest of the Ladies[***apostrophe will go here***] = Ladies' Putting Contest
The ears of the rabbit[***apostrophe will go here***] = rabbit's ears
The ears of the rabbits[***apostrophe will go here***] = rabbits' ears