I haven't really played enough games that I could think of decent rivalries I was a fan of, but I do enjoy these:
1. Scorpion and Sub-Zero (Mortal Kombat). I grew up on the '90s Mortal Kombat film and didn't really get to experience the MK games, but if I'm forced to choose a rivalry I remotely like, this is the closest one I could think of that stands out. I've seen Mortal Kombat: Legacy, and I really enjoyed the backstory behind these two's rivalry, how the tragic misunderstanding led to this eternal blood feud.
2. Master Chief and The Arbiter (Halo). Grew up on the Halo trilogy on the XBox. It was a pretty entertaining rivalry where two individuals from two species trying to kill each other became a buddy-cop kind of duo. The backstory behind the Arbiter and how that led him to team up with the MC just goes to show how rich the Halo universe can be. I kinda hope he would show up again in the later games, not that I would know since I haven't played them.
3. Kasumi and Ayane (Dead or Alive 3). DoA3 was one of my first games when I bought the XBox, specifically because I got to play it all day (while playing hooky) when it's on display at a major bookstore in Singapore. This was the Street Fighter and Mortal Kombat of my generation, so I got to know the characters' backstories pretty well, and Kasumi and Ayane's blood feud just felt so interesting, how Ayane for bound by code to kill Kasumi because the latter's a runaway shinobi. It was the first time I got to know about ninjas and their code, so it was a fascinating experience for me.
4. Red and Blue (Pokémon Red and Blue). This one isn't really appealing for me because of the rivalry and more so because of the ability to show how much Blue sucks when he loses against you at the end and got called a terrible trainer by Professor Oak. Smell ya later indeed, Blue-sor. lol The rivalry portrayed in Pokémon: Origins though doesn't appeal to me as much because I found Red to be insufferable and hypocritical. I just never really liked Red over Ash because at least Ash loved his Pokémon; Red came off like a typical Pokémon gamer who treats his Pokémon like tools (which gamers are justified in doing so, but it's hypocritical for Red to call out Blue on this when he's no better).
That's all I've got. lol