Spider-Man: The death and replacement of Peter Parker in the Ultimate Comics definitely polarized the fanbase between those who like newcomer Miles Morales and those who want him gone and Peter back.
Green Lantern. God help you if you try and discuss who is the best Green Lantern; the fight between fans rallying behind their preferred Lantern is one of the most persistent in fandom. Major factions include old-school Hal Jordan fans/Superfriends fans who feel his character was assassinated by turning him into the supervillain Parallax during the Zero Hour event, Kyle Rayner fans who feel that he's the most creative user of the Green Lantern (and that Hal was a bland character with little personality), and fans of Justice League/Justice League Unlimited cartoons who were first introduced to John Stewart and wonder why comic book fans are arguing about two white guys when Green Lantern is a black ex-Marine.
Batgirl: Fans are split over their preference for BarbaraGordon, the original Batgirl, or her Darker and Edgier replacement Cassandra Cain, or her Lighter and Softer replacement Stephanie Brown. Further complicated by the fact that Barbara Gordon fans are split between those supporting her uncrippling and those wishing it happened again. There is always been an Unpleasable Fanbase since Cassandra Cain first took the mantle, and now that 3 fairly popular and quite different characters have had the role, it gets worse, even though all 3 have had successful times under other monikers. Which meant that if DC had simply provided all 3 with a consistent role, they could probably avoid such difficulties, but replacements cause the replaced to be Put on a Bus and ignored.
Supergirl: The fanbase is polarized between fans of the original Supergirl Kara Zor-El and fans of the other Supergirls that DC spent eighteen years attempting to replace Kara with. Kara's fans feel she's the only Supergirl who actually makes sense, worked fine during twenty-six years until DC killed her off, and her death and failed attempts to replace her with short-lived substitutes led to a gigantic Continuity Snarl. Other Supergirls' fans think Matrix, Cir-El... are better characters and resent Kara's increased popularity and exposure since her return in 2004.