Yes, 15th century knights used longswords. These were adopted in part because plate armour made shields redundant; swords of 'hand and a half' length offered superior reach. They were also short enough (generally under 36") to use one handed - though not as fast when used one handed, wielded this way they seem to have been good enough for mounted men at arms attacking from horseback, since they were depicted carried by mounted men. These swords are different from truly dedicated two-handed swords (distinguished by both very long blades of 40"+ and very long grips), which were used rarely in the 15th century, though a few examples survive.

Longswords were carried at the side, much like a shorter sword would be. You can see them in a number of funerary effigies from England:

• Thomas de Motun d.1440 - this sword is cut off, but you can see the general length

• John Harington, mid 15th century

You can also see some sheathed, longer swords hanging from the hip of men at arms in Uccello's The Battle of San Romano.

In English depictions, these would be suspended from its own belt, hanging diagonally across the body, which was separate from the normal belt (worn at the waist or hips). Some depictions also show a more sophisticated suspension from the main belt.

In general, back sheaths were not used in medieval Europe - the ones that are often depicted in movies and games (where the sword is slung on the center back, with the pommel peaking over a shoulder) is not practical because it does not allow a sword to be drawn in one motion. The Japanese method of mounting a scabbard more or less horizontally around the small of the back was not used in Medieval Europe.