1. Light Yagami Versus L Lawliet (Death Note)
The rivalry between these two doesn’t necessarily need the “BRAWL” but the brains. Light Yagami uses a shinigami notebook as his power to take away peoples lives in an instant and L Lawliet is the best investigator there is who is capable of solving the crime and catching Light for his wrong doings.
2. Edward Elric and Scar (Fullmetal Alchemist Brotherhood)
Both are alchemist who is totally the opposite to each other, Scar is capable of an alchemy only to destroy,and Edward is capable of create and reshape.
3. Shizuo Heiwajima Versus Izaya Orihara (Durarara!!)
Every time these two crosses paths they just want to kill each other,and for what reason?Well I guess they got nothing more interesting to do. But their rivalry is really fun to watch especially for the damages they do like the flying objects during their fights.