I have been thinking about this for a while, as the normal way of wearing swords on foot couldn’t work that well mounted. The scabbard would slap the horse during a charge, and the way the handle sat it’d get in the way.
Looking at the bao tapestries the seem to sit further back and more straight down so the scabbard wouldn’t get in the way as much.
I know that on hussar saddles there are straps to strap the scabbard on the saddle itself, either in front or in the back, but I haven’t seen this done on any medieval saddles.
How was this done, or was swords just used so rarely in that period that the few times you needed it your squire would just hand it to you when you needed it?
Most knights wore their sword from a belt on their weak-side hip. The top of the scabbard world be about even with the belt. Since knights rode in a rather stiff, straight legged, forward foot, position the sword would hang straight down. Yes, it probably flopped around, but given the amount of armor both horse and rider would wear in battle, I doubt either noticed! The fact the scabbard attached at a single point, would allow it to be drawn fairly easily by grasping the handle of the sword with the right hand (sword on left hip), pushing it down and forward, then pulling forward.
The sword was probably the most important weapon of a knight in most battles after the late 13th century since by that period the heavy lance (14' long 30-40lbs) had replaced the lighter (overhand) lance (seen in the Bayeux Tapestry), the heavy lance could generally only make the initial strike of a charge leaving the sword as the only weapon for the melee that followed.
A knights war-saddle was very different from the latter hussar saddle. The knight's sword was straight bladed 33" to 45" long and weighed about 2.5lbs. Later cavalry swords vary in length, weight and style. Hussars normally carried a saber (curved blade) which weighed considerably less.
Drawing a sword from horseback must have not been a great issue since most mounted combatants kept carrying various types of swords for the length of time cavarly has been deployed to the battlefield. What iconography we possess shows a consistent usage of scabbards slung on the opposite side of the dominant hand - virtually in all cases, the right hand.
In the Morgan Bible from the second half of the XIII century (1250s ca.), we see a recently dismounted knight kneeling whose sword is hanging off his left hip, most likely in the same position it was during horseback travel. Plus, we see the tip of the sheath very close to the figure's ankle, suggesting it is free to move around during abrupt movements such as riding and fighting may involve.
Similarly, in the triptych of paintings depicting the Battle of San Romano of 1432 made by Paolo Uccello (1397-1475) between 1435 and 1460, the right-most mounted man-at-arms is wearing his sword on his left side while wielding a warhammer in his right hand. The painter shows a sword held on by very few connection points, possibly just two, thus possibly allowing for quite large moving space.
Aside from this, in this painting horses are not wearing armour while the fighting men are totally clad in full plate. Mounts in this period would be as heavily armoured as their riders. This surviving piece from 1480 ca. present in the Wallace Collection shows the development of plate barding for warhoses. Aside from added protection, I doubt that beasts specifically bred for warfare would feel distress from a piece of wood or leather slapping on their side.
As a footnote, swords were generally not the main tool of combat in battlefield situations. They served generally as a backup weapon when other systems (missiles, hafted weapons and so on) could not be employed. Especially in later periods, roughly from the X century onwards, shock cavalry tactics revolved around charges performed with lances whose shape would evolve according to their function, reaching a distinct shape and developing both tactics and gear pieces studied in conjuction with said tactics. Breastplates for horsemen would be equipped with special hooks under the right armpit used to lock the lance in place during a charge (named "lance rests") and a round dish-like plate over the armpit - a weak spot in the armour - to protect from skimming strikes. However, the painting of the Battle of San Romano shows two mounted warriors on the right side fighting with swords against an opponent employing a warhammer, possibly trying to aim for the armpit gap and the helmet's visor, both weak spots heavily featured in period fencing treatises regarding armoured combat.
I hope this answer helps your inquiry.
