1. Unicorns (I've edited to include bicorns)
This is actually a myth but a pervasive one and partially based on fact
In the old days there were no horse-whisperers - horses were 'broken' to the saddle.
The wild unicorn simply would not submit to being broken - it would behave like a bucking bronco in a rodeo. From exhaustion it might temporarily submit (but stand still and refuse to move forward). On recovering its strength it would be completely intractable again.
Horse breakers were predominantly men. The unicorn detects their maleness by its exquisite sense of smell. It can detect testosterone and indeed the slightest trace of semen.
In general married women found little or no success in training them, firstly because they tended to use the same methods as men - namely 'breaking'. Also because of the likely scent of semen however faint. (Hygiene standards were low in those days so people didn't tend to wash their private parts).
Virgins had neither large amounts of testosterone nor traces of semen. They also tended to be small because in those days marriage could take place at puberty.
Young girls are known to have an affinity for horses (these days manifested in My Little Pony) and are likely to tame by kindness and affection. One such little girl discovered her ability by accident when she encountered a unicorn in the forest. She felt no fear and had no concept of breaking a horse. She was able to interact with and even ride the unicorn. She ran back to tell her friends and parents and the myth arose from there. Thereafter virgins were used in any dealings with unicorns that required them to be tamed or even ridden.
In response to a comment:
"...in the old days there were no separate bedrooms. If there was a bed, it was for everybody in family. So I would hardly say that virgins had no male odor on them. – L.Dutch"
It could be that unicorns mainly frequented the royal forests because they were protected from poaching there. The only virgins that encountered them would be daughters of nobles and would likely have a separate bedchamber and handmaid if only to obey the dictum, "Children should be seen and not heard". Note: Even royal hygiene was hit and miss at that time however pre-puberty there is little B.O. of any kind.
P.S. This raises the question of whether virgin boys would qualify. Maybe they would but I suspect that they would be brought up in a male tradition of hunting or subduing forest creatures rather than being kind to them.
The Bicorn is a creature (often described as a part-panther, part-cow creature with a human-like face) that has the reputation of
devouring kind-hearted and devoted husbands
The obvious answer is that bicorns refuse to let almost anyone ride them. All they are interested in is devouring kind-hearted husbands. Maybe they beguile men to ride them and then after a suitable interview to determine whether they are kind-hearted or not, stab them with their horns in a backward movement or throw them off accordingly.