"Mad World" by Gary Jules. This song played on the episode entitled "The Bus". You may find this song on the soundtrack for the movie (done by Michael Andrews).
The original, which was done by Tears For Fears, is also on this soundtrack. "Trouble" by Coldplay. This song played on the episode "The Friendly Skies".
It was about a stewardess who disappeared. You probably recognized the actress who played the stewardess. She was Heather Donahue, who starred in "The Blair Witch Project".
"Somewhere Over The Rainbow" by Israel Kerzekeringen. "Fire Sign" by David Berkeley. This song played on the episode "Wannabe", about the boy who tried to hang himself after being taunted by his schoolmates.
This song played on one of the episodes about Graham Spaulding, the headmaster of the Spaulding Academy, a private boarding school. A student had gone missing, and Jack suspected that Spaulding was behind the disappearance. Fans often ask who actually sang the song.
It was sung by Dianne McCauley. Here's a note from CSUB Runner Entertainment (which I think misspelled McCauley's last name): "Using the same formula as "In This World," "One of These Mornings" has the same effect, but this time, Moby uses a new musical technique, in which he takes Dianne McCaulley's voice and turns it into a male's voice, but later fades it back into her own voice." That page identifies several of the vocalists and the songs they sang.
All of the guest vocalists for the album were Jennifer Price, Azure Ray, Dianne McCauley, Angie Stone, MC Lyte, Freedom Bremner, Sinead O'Connor, Shauna Lorraine Phillips, and The Shining Light Gospel Choir. "Hallelujah" by Jeff Buckley. This song played on the episode "Fallout", part 2.
is a good resource to locate songs, artists, and the episodes on which they had played. "Mad World" by Gary Jules.