Here is a overview guide about the story boos in the game: name, location, stigma and rewards.
For the first victory against each, per playthrough, you will receive a rare Amplification Stigma, with above average stat boosts. These are probably among the best ones in the game (that are easily obtainable) – some of the higher ranking World Reward Stigmas provide on par or better bonuses (Eager Defector, Rocket Dive, Entelechy, and Doomsday Repair as an example).
If you want multiple of these Stigmas, you need to NG+ the same file again and again until you have enough. You are able to start another NG+ from a file that has already completed a NG+, so theoretically you could get up to 200 of a specific Stigma.
Go-Home Club (Possible Spoilers)
[Obligatory Possible Spoiler Warning]
In a similar fashion, if you return to the Sorrow Gate in the Grand Guignol with a party of Musicians, μ wil comment that there are memories of your battles with members of the Go-Home Club. Touching this remnant will put you into a boss-rush style battle against teams made up of the various members.