I choose armor and weapons mainly for their looks. I don't use a bow because I don't like seeing the arrows on my chapters back. Or if I do I always in equip the arrows after I use my bow.

I love one handed swords, so my question is what are the swords that do not have a scabbard?

Iron sword. Nord hero/ancient nord sword. Falmer sword. Forsworn sword. Steel sword? That's all I can think of at the moment.

I want to say that glass swords have no scabbard. I imagine orcish don't either since they are so jagged.

Sorta off the original question, have you tried using conjured weapons, as you dont see them, and in the case of arrows they disappear after you put away the bow.

I think dwarven swords as well, but I may be mixing it up with Oblivion.

Daedric weapons don't have scabbards. They just get a metal ring added onto your armor that the weapon hangs from.

I think the Orcish sword has a pretty cool scabbard. I hate the way they are so rigid, they should flop about the place a bit.

Scabbards are good. I don't like how some of the npcs don't have them on some weapons that the PC does, and how the scabbard is lost when you drop the weapon.

I know glass swords don't have a scabbard, just a really cool looped belt.
Also, Dawnbreaker. I don't think it has one either.

I don't know if I would want a un-scabbered blade swinging down around my legs. Might lose something important to me.

Curved swords, CURVED swords

Aren't Blades' Swords and Imperial Swords the only swords that come with scabbards?

Steel swords come with a scabbard.

Skyforge steel has a scabbards. Blades swords have them. I don't think ebony or eleven have them. Honestly haven't even equipped an eleven sword, do find plenty laying around thanks to my one man war against the Thalmer.
But back to the question, glass does not, the nightingale blade does not, nor scimitars, orcIsh sword is open as well.
The only other one I can think of is imperial swords have them.