Namehere said:
Rome bares no relation to the armies of Europe. If it did they'd have utilised more infantry tactics and less cavalry. The Roman 'cavalry' were 'auxiliary' troops, graduates of the armed forces generally given land and levied when the army was operational around them. They were free men, Romans, who were temporarily brought back into active service and veteran forces. This was very different from European armies. For instance, are you a knight of the Catholic church? Whose horse do you think that Knight rode? Can you say, the Pope's? So no, you can not compare the Roman Empire's military structures to those of Europe. Even today there can be little comparison. One doesn't have to buy their weapons and vehicles in the modern forces and one wasn't usually expected to in the past.
Rome certainly counts. You didn't specify a time period. And while they tended to use auxiliaries as cavalry because their own was weak, the auxiliaries had also to provide their own horses.
And later knights had to provide their own horses. Maybe they also provided horses for their lance, but as those were most often foot soldiers anyway and a knight with a warhorse, a ridig horse, a pack animal and maybe a reserve horse, two squires, two archers and two speermen would certainly be one of the stronger setups. On the weak end you had one knight with horse and two footmen including qsuires. A quite common configuration had one additional light rider per knight, the rest footmen. So yes, it is save to say, that in times when knights were important, people usually used their own horses.
This became actually less common, when the big mercenary cavalry units came up.
Personal property and knightly orders where members had to "give up all properties" are kind of a special case. But in practice those still used their own horses. Either per special rule allowing to own horses, weapons and armor or by practice of gifting the horse you would use to the order. And no, the riches and land of the order is not really property of the pope.
And if we actually leave the knights behind and include eastern Europe and all those nomadic and seminomadic people who were actually famous and feared for their cavalry, those riders also owned their own horses.