I already asked this in a few topics but it was not answered and I think new topic is better for the question. So, how vulnerable horse in a close fighting that lasts some time is? Would infantryman automatically strike horse when horseman appeared in front of him or there is no time to waste on horse while man on the horse attacks you? I don't think horse charge on spearmen, of course spearmen would try to stop the charge with meeting the horses with spears. And what about cavalry vs cavalry fight? Would cavalrymen try to kill the opponents horses as soon as possible or concentrate on the real enemy?

Difficult to say. It would depend hugely on training and reach of weapons. The problem is to get close enough to the rider involves putting yourself in the way of the angry horse, so you'd probably try and stab the horse, which for most cases would not kill it instantly, and rather just get it annoyed at you. It may die later, but its definately possible you'd die before you saw the horse go down.

Wether or not one aimed at the enemy's horse in a cavalry vs cavalry fight depended on several diffrent factors. Killing the horse was seen as a dishonorable way of fighting by the French for example and even less honorboudn men viewed it as a waste of a valueable resource. War horses were expensive and therefore prefereably capture rather than killed if possible.

When fighting heavily armoured opponets killing the horse could very well give you a much needed advantage. Gustavus Adolphus instructed the Swedish cavalry to kill the enemy horses rather than making futile thrust at the heavily armoured cuirassiers.

When infantry fought cavalry the horses seems to have been the main target, bring down the horse and you wil have the man at your mercy as well. The horse would also be the largest and least well protected target as well most of the time.

Also, the chinese had a polearm which translates as "Horse Chop Knife". One assumes it was used for chopping horses...

If I was put in a situation where I was on foot, but with a group of my fellow soldiers, I'd say I'd try to take my foe off the horse, and take the horse later. I assume we're using spears/pikes/polearms etc.
If it was mano-a-mano, mounted to foot; pardon my French, but screw the mount, if I take it out, with any luck my foe will go flying and break his neck. Once again assuming I have a spear/pike etc.
If I was in a knight to knight situation (or hobliar or anyone on a horse),I deal with the guy on the opposing horse first, as A) He's worth more and B) even though war horse were trained to bite and kick and attack (I assume anway), they can't hold a sword very well :D
And then of course vise-a-versa on all the above.
Basically, I'd assume if you're on foot you'd hope you've got a pole arm, and if not, well, one could always use their shank-ponies.