Fiends

Most Fiends in the game are rare (probably 1/256 chance) high level (80+) encounters in specific areas with have great item drops. When you walk over the area, you might get a message saying a dangerous demon is nearby and will have to confirm whether you stay or not.
If you die in one of the fights, pay Charon to ensure you'll be able to fight the Fiend again. Saving after paying Charon is recommended.
Locations:
Chemtrail: by the fountain at Shinjuku West Exit Park
Matador: middle of Toyosu Shelter
White Rider: Ginza, in the area that requires Black Card
Red Rider: Kasumigaseki, in one of the emergency shelters (*requires ID Card Lv 3, behind door that requires 100 Lu to open, NEUTRAL ROUTE ONLY)
Black Rider: middle of Tennozu Shelter
Pale Rider: Minami Sunamachi, walk straight from the entrance, no need to enter poisonous area
Trumpeter: Kiccigiorgi, where the roads meet at an X
Picture: https://dl.dropboxusercontent.com/u/10886965/Fiends.jpg

The Fiends not encountered this way are:
David: Early quest from K's bar to defeat Calamity Hordes.
Mother Harlot: NG+ Chaos sidequest.
4 riders battle: After the alignment lock, return to Infernal Tokyo (Chaos and Neutral only), talk to demons and you'll receive this quest. Finishing this quest does not unlock their fusion.