OK, this is a list both compiled from a post from silktail and my own discoveries. (Thank you silktail for your original post)
All songs appear between 10pm and midnight. They don't all appear at 10pm, sometimes the person has to walk to their location and can show up late (@11 for example).
1. Next to grunt house outside castle 2.Farmland, behind west merchant shops 3. Hidden Grove by art dealer 4. Next to the Hot Spring to the N.E. of Castle Town. 5. Behind the tree next to Skinny Ray's classroom and the flying machine. (Farmlands.) 6. Next to river near Glamour Town bridge 7. On the north side of Soup Church, in Castle Town. 8. Just south of the eastern Farmhouse in Farmland. 9. Apple Town, in corner next to river (where gem was found) 10. By the Knight statue in Solder Town. 11. In front of house built over water in grassland town 12. Under the left lamppost, next to the Gourmet Town sign. 13. Along the wall east of the eastern Stonecutting Plant in Stone City. 14. On the east side of the Red roof, kebab house in Gourmet Town. 15. Behind the Cow Shed in Farmland. 16. In the inside corner of the L-shaped (clothing) house, which is next to Solder Town's cannon. 17. East of Soldier Town cannon (between tree’s) 18. Next to stairs in soldier town that lead to overlook of the marble dragon pen. 19. Behind the tree north of the King's Natural Reserve gate. (Behind a Glamour Town mansion.) 20. Behind Animal Hunter House 21. In the passage south of the mansion west of the hospital, south of the theater. (Glamour Town.) 22. By the tree behind the Hospital in Glamour Town. 23. Between the Eggan Civilisation Ruins' gate and the Stonecutting Plant in Miner's Town. 24. Near mine @ gate to TV land 25. Royal Town, next to N. Side of school. 26. On the sand of Over there Beach. 27. Wizard Town, near bridge next to river (not thru gate) 28. Atop the Drain Cover on the N.w. side of Royal City. 29. On the N.W. side of the Theater in Glamour Town. 30. Final tune quest after collecting the other 29 tunes. (Alpoko Castle 5-7pm?)
Note that they won't appear every night, you might have to wait several days before they appear in a location. You may also need to go to bed and wake up the next night to reset the songs if you can't find anything after several tries.
Once the songs are found, the people walk around singing them and multiple people can sing the same song. A person with an undiscovered song stands still and you can't hear it (it sounds like they are talking, not singing) !! Sometimes a wandering person will stop and then sing (evidenced by notes).