The right side....and I can tell you why (you'll like this).
Most people are right handed, so would wield their sword in their right hand....as I'm sure you do. The scabard would have be on their left side (try it with your imaginary sword). So to mount avoiding getting your scabard caught up, you have to get on from the left.
And you want to see the conformation of the horse you are mounting, obviously, to check that it will carry you safely in your battle...and to see the conformation of the neck, the mane must be tucked away on the right.
Voila!
S
right side................wish some one would tell my mare though!!!!.................no matter what i do it allways falls to the left!!......never had that problem with her foal hers grew and lay to the right without any interferance from me!!!!!!