I think that the Sword/White Sword/Magical Sword are iconic and I've always had a special place in my heart for them, but I've never liked that they essentially disappeared from the series and weren't given any backstory.
The Koholint Sword is one of my favorites as a reward for collecting Seashells. Much more satisfying and intriguing than Korok poop.
The Gilded Sword is also one of my favorites, as its forging is very linked to the time cycle mechanic.
The Phantom Sword is my favorite of these three, for it is the only blade that can directly control time. I love Zauz and that Link reunites the World of the Ocean King through gathering the materials necessary to forge the blade, and then combines it with the Phantom Hourglass itself, giving him the ability to stop time. Like, badass.
But my favorite has to be the Four Sword. Link's development of it throughout Minish Cap, working towards being able to split himself, and then the great gameplay it provides in Four Swords Adventures, and then its return in A Link to the Past + FS, where it is used against a new generation of Link is both a compelling story, and a great gameplay mechanic.