Content Warning: This list contains a mention of sexual assault
There are the battles between heroes and villains that have fans cheering. They want to see Superman defeat Lex Luthor or Wonder Woman destroy Ares. These are the characters that deserve their demise most of the time thus making the fans root for the hero to save the day.
In the more tragic scenario, there are battles that break fans' hearts. Sometimes the hero has to fight a villain who turns out to be someone they care about. A friend, a lover, a family member: these are the fights that leave fans divided internally and wishing that things ended differently.
Batman & Catwoman
A result of one of the most toxic relationships of the DC universe. So many times, Catwoman and Batman fall for each other but something bad always happens. She tries to give up the life of a thief but relapses. They attempt to get married and it turns out to be one giant fakeout.
Batman is no saint in the relationship either; he is almost zealous when it comes to his mission as Batman. His hard shell is nearly impossible to penetrate, even with the woman he seems to genuinely love. In the end, the two always end up returning to being a vigilante fighting a talented cat-burglar.
Starfire & Blackfire
Two sisters with different ideologies end up becoming mortal enemies. Koriand'r A.K.A. Starfire values life, peace, and justice in the universe while Blackfire sees things in a darker light. Blackfire is much more murderous and with a ruthless dictator mentality.
On top of that, despite being the eldest sister, her unstable nature led to the younger sister Starfire becoming the heir to their people's throne. Of course, this sets her off and turns her into Starfire's nemesis. Since then, the two have shown to have cared for each other but something always splits them apart.
Thomas Wayne & Martha Wayne (Flashpoint)
In an alternate universe, it's young Bruce Wayne who is killed in Crime Alley, leaving Thomas and Martha alive. This sends the grieving parents on two very different paths: Martha devolving into a cackling maniac as that world's Joker. Thomas Wayne becomes a gun-toting version of Batman.
Thomas Wayne's Batman is much darker and unhinged but still fights crime, trying to stop his own wife. Nobody can blame Martha for losing her mind after losing her only son. In the time these two are shown together in Flashpoint Paradox, this hero/villain relationship is just tragic in every way.
Harley Quinn & Poison Ivy
Everyone's favorite jester has reformed to become a surprisingly engaging anti-hero. She has also developed a relationship with Poison Ivy, a much superior relationship after the Joker. However, that doesn't mean that it's a perfect relationship between the two.
Sure, the two ladies bring out much of the best in each other and are adorable together. Unfortunately, both Harley and Ivy suffer from mental issues. Poison Ivy's radical eco-terrorist ways are not always what Harley agrees with. And Harley, sadly, always falls for the Joker thus splitting the two apart.
Batman & Ra's Al Ghul
As much as Batman and Ra's Al Ghul will battle each other, they have equal respect for one another. Ra's is a genocidal maniac but he is also an intelligent gentleman with some wholesome moments in his life. The two are not quick to instant get into a fistfight, they will talk and there are even times where Batman and Ra's will agree to a truce.
Unfortunately, that is not always the case. Ra's Al Ghul is also obsessed with destroying the world to restart it or trying to make Batman take his place as leader of the League Of Assassins.
Nightwing & Tarantula
This friendship ended up becoming one of the most depressing ones. Tarantula is a villain whom Dick Grayson saw potential in. He even managed to take her under his wing and tried to turn her into the newest hero. At first, her reforms seem to work as they become lovers and partners.
However, when Tarantula relapsed, she relapsed hard. Not only is she still a murderous criminal but during a state of shock, she raped Nightwing in one of the most controversial sequences in DC comics history.
Batman & Red Hood
Batman makes a lot of enemies out of his friends. In this case, it's his own adopted son Jason Todd AKA the second Robin. Jason was already an angry and violent person, to begin with. However, after Batman failed to save him from being bludgeoned to death by Joker with a crowbar, it sent him over the edge.
Jason returned to Gotham as the Red Hood, a vigilante/crimelord that sought to make Joker suffer like he did while taking control of Gotham's criminal underworld. This, sadly, makes him the enemy of Batman leading to an epic confrontation between them. Fortunately, Jason Todd has shown that he can reform and act as a vigilante again in Gotham despite the urge to kill. He even swears off killing at times, such as in the Gotham Knights game.
Wonder Woman & Cheetah
For this, it will depend on the universe and who is writing the characters. However, there have been times where Diana Prince and Barbara Minerva are shown to have once been friends. This falls apart when Minerva is given the Cheetah's curse and becomes the bloodthirsty rival of Wonder Woman. This was the case for Cheetah's live-action debut in Wonder Woman 1984.
Even as a hero and villain, the two still show an emotional connection to each other. In some cases, it's hinted that the two lust for each other and if it weren't for their two different paths, they would have evolved to become a couple.
Green Lantern & Star Sapphire
Another lover's quarrel but on a more galactic scale. Carol Ferris, Hal Jordan's lover, became possessed by the spirit of the Zamaron queen. This granted her a pink ring that gifted her the power of love but also made her an adversary of the Green Lantern Corps.
Since then, Carol and Hal Jordan have battled many times. They have also been allies and lovers who have a relationship as stable as Harley and Joker's relationship. However, Star Sapphire has reformed more often than Joker ever will.
Batman & Superman (Injustice)
In the Injustice series, Batman and Superman became total opposites. Superman became a ruthless dictator and murderous psychopath while Batman led the Insurgency against him. The two have as much hatred for each other as they still have that brotherly connection.
Even in Injustice 2, the two work together to stop Brainiac only to fall back into conflict. Batman and Superman have a heart-to-heart talk about how they both wish they could go back to the way things were. Unfortunately, Superman is far too evil and the two will likely forever remain enemies.