"In Dreams" and "Into the West." Though I've really liked most of the songs in the new Hobbit movies, with the exception of the Goblin King's, which seemed to syncopated to be very Tolkien-like.
The Lay of Nimrodel never appealed to me greatly in the books, but I love this version.
Also the Lay of Boromir is amazing, and I like this rendition a lot.
It seems like a lot of people who try to put the songs to music make them too classical-sounding. They're ballads, people! Nothing against classical music, but the point of ballads is words, not music. Get too musically creative and you murder the thing by accident.
Bridge of Khazad Dum always grabs me but Breaking of the Fellowship will always remain my fav soundtrack. It was already back then and it still is today
He makes Heavy Mithril versions of Howard Shore's themes. They're amazing. And on the subject of Heavy Mithril, Blind Guardians's "Nightfall in Middle Earth" album is actually based on The Silmarillion and not LotR, but it is excellent.