Which rivalry is the best?

I would say that from American League to National league, the biggest rivalry would be the Red Sox and Phillies, especially now, as they are on the top of their respective leagues.

Tough call. Historically, I would say some of the best of all time were:

NY Giants / Philadelphia Athletics

NY Yankees / Brooklyn-LA Dodgers

St. Louis Cardinals / Chicago Cubs (doesn't apply to your question today, as they are in the same division, but dating back to before divisions, it was still a great rivalry).

In the more modern era, I would say St. Louis / NY Mets. They had some great battles in the 1980s, which was revisited in 2000 and 2006.

Mets - Cardinals.

When the Dodgers were in Brooklyn, their biggest rivalry other than with the NY Giants was with St. Louis, because of pennant races with them in the 40s, and the tape measure HRs that Musial used to hit in Ebbets Field. After the move to CA, and the Mets entering the league in 1962, their next biggest rival (other than vs. Giants and Dodgers) was with St Louis.

This peaked in 1985, 1986, and 1987, and again in 2000 and 2006, and I heard Ozzie Smith and Keith Hernandez once say that this rival actually raised each team to a higher level, and, as a Mets fan, in my opinion, it was better than any other NL rivalry in the last 30 years. The Dodgers- Giants rivalry in CA is a watered down version of what it was in NY in the 20's through the 50's, and I don't see the Cubs-Cardinals as a major rivalry.

Phillies vs Yankees.

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Giants and Yankees have a historic rivalry, dating back to when the Giants were still in New York. But, I'd have to say Giants-A's. A rivalry without violence. There you go.

Battle of the Bay Giants vs A's

Red Sox vs. Indians. They played bean ball last year and might do it again in the future.

White sox vs cubs

Red sox vs angels

Mets vs yankees

Phillies vs yankees

Cardinals/Mets, people from New York still don't want to admit that Yadier Molina beat them in game 7.