NiTessine
Explorer
Well... I'll take a classical music viewpoint. Partly because I do not know anything composed sooner than 50 years ago, and partly because I despise modern music. 
Barbarian: "Ride of the Valkyries" from Richard Wagner's opera Valkyrie.
Bard: The aria "Largo al Factotum" from Gioacchino Rossini's opera "Barber of Seville".
Cleric: A short bit from Mozart's Requiem could do it. Goes for necromancers, too. Alternatively, pick any Gregorian chant.
Druid: Part four of Beethoven's Pastoral Symphony, "The Storm".
Fighter: "Mars, the Bringer of War" from Gustav Holst's Planets.
Monk: "Wrath of the Yellow River", a piano concerto by Xian Xinghai.
Paladin: Overture from Wagner's "Tannhäuser".
Ranger: "Sounds of the Forest" (or something like that) from Wagner's Siegfried.
Rogue: "Carefully on Tiptoe Stealing" from the Gilbert & Sullivan operetta H.M.S. Pinafore.
Sorcerer: "Uranus, the Magician" from Gustav Holst's Planets.
Wizard: "Wizard's Apprentice", by Paul Dukas. You know, the Mickey Mouse thing from Disney's Fantasia.

Barbarian: "Ride of the Valkyries" from Richard Wagner's opera Valkyrie.
Bard: The aria "Largo al Factotum" from Gioacchino Rossini's opera "Barber of Seville".
Cleric: A short bit from Mozart's Requiem could do it. Goes for necromancers, too. Alternatively, pick any Gregorian chant.
Druid: Part four of Beethoven's Pastoral Symphony, "The Storm".
Fighter: "Mars, the Bringer of War" from Gustav Holst's Planets.
Monk: "Wrath of the Yellow River", a piano concerto by Xian Xinghai.
Paladin: Overture from Wagner's "Tannhäuser".
Ranger: "Sounds of the Forest" (or something like that) from Wagner's Siegfried.
Rogue: "Carefully on Tiptoe Stealing" from the Gilbert & Sullivan operetta H.M.S. Pinafore.
Sorcerer: "Uranus, the Magician" from Gustav Holst's Planets.
Wizard: "Wizard's Apprentice", by Paul Dukas. You know, the Mickey Mouse thing from Disney's Fantasia.