My question: There are three tunes in 3D Lemmings which play on only one level each. What are the names of these three levels?
No, there are none. (Unless you don't count practice levels, in which case, Tricky level "Garden Maze"). Here, for each music, is at least two levels in which they are used(most have more, I'll point out if they don't):
Space Music 1: "Alilemms"(Fun) and "Lemlab"(Tricky)
Space Music 2: "Dot To Dot"(Tricky) and "Rescue Team"(Taxing) <--- These are the only two
Army Music 1: "Take A Dive"(Fun) and "Sliding Maze"(Fun)
Army Music 2: "Alpine Assault Course"(Fun) and "Let The Race Begin"(Taxing)
Circus Music 1: "Bounce Bounce"(Fun) and "T'uther Circus Level"(Mayhem)
Circus Music 2: "Which Exit?"(Tricky) and "Stilt Walker"(Tricky) <--- These are the only two
Computer Music 1: "Lemmings Inside"(Fun) and "Lemmtris"(Taxing)
Computer Music 2: "Breakout"(Tricky) and "Motherboard"(Taxing)
Pyramid Music 1: "Tooten Lemming"(Tricky) and "The Arena"(Taxing)
Pyramid Music 2: "King Coder's Tomb"(Tricky) and "Raiders of the Lost Lemming"(Mayhem) <--- These are the only two
Maze Music 1: "The M-A-Z-E"(Fun) and "Final Maze"(Mayhem)
Maze Music 2: "Garden Maze"(Tricky), and many of the Practice levels. If you don't count these, Garden Maze is the only level.
Golf Music 1: "Hole In Ten"(Fun) and "Land Ahoy"(Tricky)
Golf Music 2: "Fore"(Tricky) and "Driving Range"(Mayhem) <--- These are the only two, *I think*
Lemgo Music 1: "Attack of the 50ft Lemmings"(Fun) and "A Head Above The Rest"(Tricky)
Lemgo Music 2: "That's Right"(Fun) and "Play Time"(Tricky)
Castle Music 1: "It's A Run Around"(Fun) and "The Five Arches"(Mayhem)
Castle Music 2: "Castle Lemalot"(Fun) and "Catacombs"(Mayhem)
Candy Music 1: "The Bean Machine"(Fun) and "Candyland Climber"(Fun)
Candy Music 2: "Chocolate Drop"(Tricky) and "Jelly Belly Islands"(Mayhem)
EDIT: This is assuming you use CD Music. I'm not sure about otherwise.