Reprinted from Mythras Core for convenience
Unless learned as part of specifically mounted combat styles (see Combat Style Benefits page 88), the combat skills of a rider may not exceed their Ride skill. The advantages of being mounted are considerable however:
A mounted warrior rolls a 1D10+10 for the Hit Location of bipedal creatures, provided the target is lower than the rider.
A mounted warrior can withdraw freely from engagement on his turn provided his mount moves faster (i.e. has a higher Movement value) than the opponent, and the mount itself is not engaged in attacking. This does not mean they don't have to spend an Action Point, but it does mean they don't have to make an opposed roll to withdraw
A mounted warrior may, at the cost of an Action Point, substitute either his own Combat Style or Ride skill instead of his mount’s, to defend it against attacks.
A mounted warrior may combine his SIZ with his mount’s for the purposes of avoiding Knockback, provided he passes an unopposed Ride roll.
A mounted warrior may, when charging with a braced weapon, substitute his own Damage Modifier for that of his mount. A weapon braced for a charging attack cannot be used to parry. If the weapon impales it must be left in the body of the target, or else break or dismount the rider as the mount continues past.
Warhorses, by definition, are trained with the Attack trick (Classic Fantasy, p71), though not normally versus undead. They may or may not be trained with other tricks.
Draft horses normally learn the Work trick. They are not often taught other tricks, but are certainly capable of learning them.
The rules for the character attacking while mounted are covered in other areas, but are generally capped by the Ride skill unless a trait, feat, or class feature bypasses this
A mount trained with the Attack can be made to attack while you are mounted with a simple command. This takes an action from you, and the mount attacks with its own combat style skill. Normally this is capped by your Ride skill; However, you can choose to allow the horse to attack more freely and ignore this cap. In this latter case, any attack that is over the character’s ride skill, success or failure, forces the character to make an unopposed Ride check or fall off the horse and go prone.
You cannot have the horse attack more than it has action points. Generally a mount is restricted to attacking in it’s front and/or rear facing. A horse, for example, can attack either the front (with a traditional rear and plunge) or the rear (with a kick)
Warhorses called by the paladin ability come having learned the Attack trick (against living and undead) as well as others up to the maximum allowed by their INS.
A paladin may invest experience rolls in his warhorse. If the paladin’s warhorse is killed and he wishes to summon a new one, this new warhorse comes with the experience spent in the previous one.
At rank 1, Cavaliers do not automatically gain warhorses, though many acquire them very soon thereafter. While they have great skill on a mount, their mounts are not magically summoned, and possess normal traits for the species of the mount. As befits their station, they tend to seek high quality mounts with great training.
If a cavalier's mount dies, that does not guarantee a new mount is available. It'll happen, probably. It does not necessarily have the same training, but this is compensated for by the buying up of the oath and the greater abilities of the mounts.
Pegasus are not detailed in either Classic Fantasy or Mythras, but I'll make up something.