You know, because I have way too much time on my hands, I went through the character list and looked for edgyness. The following is my opinions on the characters with dialog, and whether or not I find them edgy. Please note that this is only my opinion, and not the opinions of the Reborn development team.
Cain: Ok, so his name might be a little edgy, since it sort of reminds me of everyone's least favorite "Shadowmancer," but really Cain is not edgy. He's loud and flamboyant, but not edgy. I suppose his hair and clothes might be kind of edgy, but I don't consider him to be edgy.
Fern: I hate this guy, but he's more into "swag" than he is into being edgy. He's evil, and he loves it.
Victoria: Another character who looks more edgy than anything else, you could really just say she's more Gothic than anything.
Adrienn: While I don't really remember Adrienn, I wouldn't say Adrienn is edgy. The whole restoration project also seems to throw Adrienn into the no edgy pile.
Amaria: Sure, she probably had the worst reaction possible to the diary incident, but I wouldn't call her edgy. She seemed way too cherrful for that.
Aya: Ok, you might have me here, Aya might be kind of edgy, but since when was punk edgy? I guess I'll have to coincided though,,, Uninteresting Venam is edgy. One point for edgy.
Charlotte: She seems more crazy than anything, but dresses too brightly for me to consider her edgy.
Ciel: No, not edgy.
Corey: At first glance, one might be able to consider Corey edgy, but I feel like his character runs deeer than that. After he lost his house wife, I could see him being more depressed and desperate more than anything. Plus, he's apparently based off of a real person (the details are hazy to me) so yeah...
Florinia: Can robots be edgy? I think not.
Hardy: Seriously though, is punk edgy?
Julia: I think Julia might be the exact opposite of edgy. She's bright, energetic, and is a cheerleader. The three words that vanquish edgyness!
Kiki: Zen is not edgy. It really isn't.
Luna: Another character on the fringe of edgyness, I personally think she's more Gothic than anything.
Noel: (Man his portait makes him look like he needs to take a nap.)
Radomus: Is the Mad Hatter edgy? Maybe for Tim Burton, but not here. He's just... Mad?
Samson: "Now I'm a jerk and everyone loves me!"
Saphira: Yeah... Most of the characters who resort to murder are edgy in this game, so here's another edgy character.
Serra: Can moms be edgy? I sure hope not.
Shade: I really hate to say this, but Shade might be edgy. His lack of dialog makes it hard to determine, but he feels like a CreepySpaghetti, which are often edgelord conventions. I do love his design though.
Shelly: Someone give this poor kid a hug already!
Terra: I hate this character, but at least she's "random" enough to not be edgy.
Titania: Little miss murder is edgy, but for a steel type gym leader, she kind of needs to be.
Anna: Kind of creepy, but not edgy.
Arclight: Would an edgy character give you a lucky egg for helping stop a smuggler?
Bennett: Creepy for different reasons, but still not edgy.
Blake: He's more "MLG" than edgy, so no.
Cal: I'll need some input for Cal before I can call him edgy or not. He seems regretful, so for now I'm going to say no.
El: I almost made a GoT joke here, but to keep this spoiler free, I won't. He isn't edgy though.
Heather: I don't think so?
Laura: I honestly don't know or remember this character. For the sake of this list though, I'm going to label her not edgy.
Sigmund: Mad scientists are not edgy. They're just crazy.
The Bad Guys: One is like Two Face from Batman, and not edgy. One has daddy issues, but is not edgy. One I can't remember his personality, but looks kind of edgy so I'll count him as half. One is legitimately evil, but doesn't seem edgy.
So my final count... We have... Thirty no edgy characters to four and a half edgy characters with one character undecided. The game is dark, the plot is dark, but the characters are overwhelmingly not edgy. Feel free to correct me if I'm wrong though, this is all my opinion.