All we've seen from Azula outside the main series is from the comics The Search (2013) and Smoke and Shadow (2016).
Azula went with Team Avatar to search for her and Zuko's mother, they found her, and she vanished. She was still... unstable then. Later she formed a "fear group", fueling their resentment against Fire Lord Zuko, trying to make him authoritarian and feared. Didn't work, and she vanished again.
The Search (2013)
Azula went into what passes for a psychiatric hospital, or, as she calls it, "a nuthouse". It was certainly not in anyone's mind to let her out, but after the war had ended, Team Avatar was given a lecture on ruling, how family is important, etc - and that got Zuko to think about how he put his father in prison, his sister in confinement, and never heard back about his mother.
Zuko decided to go on a search for his mother, which is the central plot of the Avatar comic The Search (2013). As he couldn't get any info from his depowered father, he had Azula talk with him. And while she did get some info, she also broke from her restraints and struck a deal with him: she accompanies Team Avatar in the search for Zuko's mother, in exchange for the info.
They (Zuko, Aang, Katara, Sokka, Momo, Appa, Azula) traveled to Ursa's (Zuko's mother) birthplace, a small town. Unsurprisingly, the cohabitation with Azula wasn't very pleasant, especially since she still had hallucinations of her mother "conspiring" against her. Unknown to the others, she had laid her hands on a letter from Ursa that she thought could overthrow Zuko.
See, Ursa was originally from Hira'a, a small town, and she had a lover there, named Ikem. But she was married more or less by force to Fire Lord Ozai and had to reject Ikem - but she still wrote to him, and one of those letters (the one Azula obtained) seemingly stated that Zuko was Ikem's son, not Ozai's, and thus had no claim to be Fire Lord.
They get to Hira'a, and are hosted by a nice couple of actors, Noren and Noriko, and their 6-years-old (around that age) daughter Kiyi. They tell them Ursa had to got to the capital, and Ikem wandered into the "Forgetful Valley", never to be seen again. Team Avatar decides to go investigate there.
Turns out, a spirit lives there, the Mother of Faces, who has the power to give people new faces. Team Avatar meets Misu, an old woman seeking to meet this spirit to "cure" her brother, whose face was disfigured. The spirit granted them one wish, Aang left it for Misu, as she had been waiting for so long. Azula blew it (figuratively) by going first and asking what happened to her mother.
Long story short, Ursa had a new face given to her by the Mother of Faces, namely... Noriko, the woman who received Team Avatar in Hira'a, and Noren was actually Ikem. She also forgot about her children, Zuko and Azula. Azula sprinted back to the town, hellbent on killing her (as she was still crazy and convinced her mother was plotting against her). Zuko and Sokka thwarted that plan, and Azula vanished into Forgetful Valley. Zuko tried to have her stay, saying he could help her, but she was still unstable and thought him too weak.
Smoke and Shadow (2016)
This happens after The Search, and Ursa, Ikem and Kiyi are in the capital alongside Zuko. It's revealed that Forgetful Valley was searched for weeks, but no sign of Azula was found. In Smoke and Shadow, Zuko has to deal with a rising anti-Zuko movement, namely the "New Ozai Society", who doesn't recognize his authority. They're led by Ukano, Mai's father.
Attempted coups and attacks on his family he could handle, but a new threat is rising, the Kemurikage, dark spirit beings who seem to boss Ukano, and eventually start kidnapping the capital's children, including Tom-Tom (Mai's brother) and Kiyi (Zuko's newly found young sister). As spirits are involved, Zuko asks for the Avatar's help.
Fear grows in the streets as more children are kidnapped. While Zuko is investigating the Kemurikage legend, Ukano gathers a militia and starts a curfew. The "police force" takes on the streets and begins fighting against the Kemurikage - except at the same time, Aang discovers that the actual Kemurikage spirit stopped coming to the human world long ago. Meaning, the kidnappers are fake.
Zuko disapproves of the militia, and that's seen as him being a weak Fire Lord not doing what's necessary to protect his people. When rushing against a Kemurikage kidnapping Kiyi, the "spirit" escapes, shooting electricity at him... Signature move of Azula. It's revealed she went back to the psychiatric institution, freed the other patients, and got them into the Kemurikage setting.
That puts Zuko a little bit on the brink, and he decides to seal off the capital, and go against members of the New Ozai Society by searching citizen's homes for clues. In doing that, he's "doing it the hard way", which certainly does not sit well with Aang, who starts his own investigation and finds a secret passage in the castle, where Azula likely went.
He brings Zuko there, and they find out that the passage lead to a graveyard where they meet Kemurikage fakers. Fight ensues, Azula leads Zuko to a crypt and reveals she's done wanting to be Fire Lord. Instead, she'd rather make Zuko into a "strong" Fire Lord, ruling ruthlessly. She escapes in smoke.
The kidnapped children are freed and Zuko addresses his people, presenting his excuses and asking for forgiveness. Azula and her companions are watching from a rooftop, Azula states that the whole speech is "touching" (with all the snark she can muster), and vanishes for a second time.
As far as I know, she's not in any other comic, nor Legend of Korra. Thus, it's not known what happened to her afterwards.