The largest current rivalry is probably between the SALE (Syndicate) and CIS (Singularity) spheres. It's been going on for quite a while now and three of the last four (4/5?) major wars have been either completely centered around or included the conflict between these two, these being Darkest Hour, Dodge This, and Global War 30 (some will split that last one into two, I don't think that makes too much sense). You can find pretty much everything I say here on the wiki with Declarations of War and other drama linked.
Unless it's longer than I know, the conflict dates back to when t$ cut off Cataclysm (who merged with Paradise to form Singularity) when Cata and Paradise hit Midgard (old bloc that had Aurora (m. w/ CTO to become EVH) and The Immortals). Claiming that their allies went in without their agreement, t$ gave Cata their 72 hour grace period before the treaty would be cut. However during that time, Eclipse hit these two alliances in defense of Midgard, and t$ did not defend Cata on the basis that they had violated their treaty by going in without t$' approval. Once the war between Cata-Para and Midgard+Eclipse was over, t$ signed Eclipse. It was a pretty controversial move for most people, and was the CB for Cata's entry into Darkest Hour, where Eclipse and t$ got rolled. It's been a state of perma-war since then I believe, and both sides accuse the other of maintaining that state. Some people may disagree with this POV, I'm an ex-t$ member that tries to stay on good terms with them still, you can ask SIN members for theirs if you wish. Overall the rivalry has gone on for too long imo, it's getting pretty boring, @FA please hug each other immediately.