Oh, I am certain that a saddle was used. I was just wondering how they didn't notice Ian grabbed it while entering Edoras. :)
I saw the white line too. I suspect it was a rope halter of some sort and I saw it during the scene when Gandalf and the others were stopping at the crest that overlooks Edoras. The angle covers Shadowfax with the bodies of the other two horses and their riders. I am certain that it was done to cover the obvious saddle lines and necessary bridle at that point (since Ian was atop him).
With all that said - what a horse! Very handsome indeed. He certainly had the attitude of a King, didn't he? Talk about perfect casting! :)
I saw the strategic placement of Legolas's horse riding up to the gate at Edoras... ;~D As for the neck rope, you can control a willing horse with a neck rope... I have done it. Did it a lot as a reckless teenager! A rope around the base of the neck can be used to turn and stop a horse that wants to cooperate. They probably used it in training and rehearsal, and removed it whenever they could for scenes where the horse could do the work without it. One thing about horses is, once you have repeated something a few times... they will anticipate cues and do the same thing again. Patterns are very easy to teach a horse. So some of the liberty shots were undoubtedly Shadowfax actually knowing the part.