The "Vileness", as termed by Cadsuane, after the Aiel War probably refers to a number of things. For starters, there was the sudden death of the Amyrlin, Tamra Ospenya, and a rash of deaths among the Aes Sedai, including several prominent, high ranking ones, most under mysterious circumstances. Meanwhile, below the surface, several other things were occurring, most of which are very troubling. There was a sudden increase in deaths among men and boys who seemed "lucky." Also, a number of channeling men were gentled by the Red Ajah illegally, on the spot, and not within Tar Valon as prescribed by Tower law.
This unsettled period in Tower politics seemed to end when the three Red Sitters in the Hall were summarily exiled, and the reasons for their removals were "Sealed to the Flame," which in effect classified the whole affair as for the Amyrlin's eyes only (although "Sealed to the Flame" can also involve the Amyrlin swearing a sister into her confidence, as Elaida did when she recruited Seaine to seek out the Black Ajah). All in all, it was a messy, upsetting time among the Aes Sedai, and all the above-described events seemed to be related.
Most of the links between the events were given out in various places throughout the series, but Legends: New Spring finally nailed down some of the events.
This is now what seems to be the sequence of events:
Tamra Ospenya, the Amyrlin during the Aiel War, was murdered by Jarna Malari, a publicly Gray sister, in an effort to learn what Tamra knew about the Second Coming [ACOS: Prologue, Lightnings, 42]. Jarna was the leader of the Black Ajah at the time. Tamra was the Amyrlin who was present when Gitara Moroso had her Foretelling about the Rebirth.
Tamra was canny enough to know that there was some Black Ajah activity afoot and swore Siuan and Moiraine to secrecy. From L:NS, we learn that Tamra really wasn't stupid enough to just send two newly raised Accepted (Moiraine and Siuan) out to find the Dragon Reborn. She had very carefully and surreptitiously called in and sent out a group of Searchers, most of whom were later killed by the Black Ajah. It is still unclear whether Cadsuane was among the Searchers, but it's quite possible she was. The ones that Siuan and Moiraine were aware of were: Aisha, Kerene Nagashi, Valera, Ludice and Meilyn Arganya [L:NS, 668]. (Side note: Cadsuane gave Kerene and Meilyn as examples of the strongest in the Tower.)
From her interrogation of Tamra before killing her, Jarna learned something about the Second Coming, but misinterpreted it, and didn't realize that he had just been born. Possibly she construed what she had learned as the Dragon Reborn was ready to announce himself, or maybe she just learned that he was alive, but had no idea of his age. It's unlikely, therefore, that any of the Searchers chosen by Tamra belonged to the Black Ajah, because then the Black sisters would have known the same details Moiraine did (i.e. the Dragon was a baby, born during the final battle of the Aiel War, on Dragonmount). They were searching blindly.
In any event, the race to find the Dragon had begun, and the Black Ajah unleashed a campaign of murder to get to him first, killing anyone, man or boy, rumored to be "lucky," on the assumption that any man who seemed lucky might be channeling, since luckiness is an outward signal of unconscious channeling [L:NS, 712].
However, while that was going on, and for a while afterwards, the Red Ajah was running a second front of the campaign. Later on, Jarna Malari became Keeper to Tamra's successor, Sierin Vayu, (the Gray with more than a touch of Red in her). Jarna, still leading the Black Ajah, implemented a program of search and destroy. She directed Galina Casban, who led the Red Ajah, to use her Red minions to seek out any man that could channel and gentle him on the spot. It might have been the work of a selected cadre of Red Sisters who wouldn't balk at this flagrant violation of Tower law. We know it is illegal to gentle a man "extra-judiciously," away from Tower [ACOS: Prologue, Lightnings, 22]. By doing this, Jarna began a process that would circumvent the Tower's "Dragon-finding Process," which was to bring every channeling man to the Tower, where they would put him to some sort of inquest to determine if he was the Dragon Reborn, and then gentle him. One assumes that they would not have gentled the Dragon Reborn once they actually found him.
One of the victims of this 'search and destroy' directive was Thom's nephew Owyn. Presumably, Owyn is among the gentled men who do not appear in the Tower records Elaida and Alviarin are discussing in ACOS. Elaida and Alviarin later discuss that merely knowing that there are channeling men who do not appear in the records is dangerous. The danger likely stems from the notion that because they are not listed means that you would only know from personal knowledge, which would suggest some involvement. Elaida's comments seem to imply that she participated in at least one of those missions [ACOS: Prologue, Lightnings, 22], and Toveine confirms this [TPOD: 26, The Extra Bit, 513].
Meanwhile, all hell breaks loose. Ishamael puts Jarna Malari to death in a derelict ter'angreal for messing with the program. We know that Ishy had notions of finding the nascent Dragon Reborn intact so that he could turn him to the Dark Side of the Force. What Jarna had done was severely reduce the odds of that happening, so it seems that he killed her for acting as a loose cannon. It is likely that this is what ended the campaign of murder by the Black Ajah, but the Reds' illegal search and destroy mission continued [ACOS: Prologue, Lightnings, 42].
Two years later Sierin Vayu died. Ishamael clamped down on the Great Council of the Black Ajah to determine that none of them had anything to do with Sierin's death. It is possible that she gave some sort of tacit approval toward the 'search and destroy sorties' (she was pro-Red) and the Red Ajah had a hand in killing her to silence her. It is also a very real possibility that Sierin Vayu herself was Black Ajah.
Alviarin seems to think that the Reds did have a hand in her death, and disavowed any Black participation [ACOS: Prologue, Lightnings, 42]. In WH, we learn that Chesmal Emry, one of Liandrin's original coven of Black Ajah, is very proud of the fact that she induced the Red Ajah into murdering Sierin, which is interesting because this somewhat contradicts Alviarin, since the Black Ajah did have something to do with the death, but didn't actually do the deed themselves. [WH: 10, A Plan Succeeds, 242] However, as John Hamby suggests, if Sierin was Black, then Galina could have used the Reds to silence her, to protect Black Ajah secrecy.
Whoever was responsible, it was to no avail. The secret came out, and the Red Sitters in the Hall were exiled. Perhaps the rest of the Hall assumed that they had also exiled the Red Generalissima, because many Ajah heads also hold Seats in the Hall, or it is possible that they realized that they could never learn who really led the Red Ajah, and enacted their justice on the high ranking Reds they had at hand. Thus they exiled Toveine, Tsutama and Lirene, but left Galina untouched. We don't know at this time whether the "Red Purge" occured under Sierin Vayu, or the Blue Sister, Marith Jaen, who succeeded her.
There is a slight quibble regarding the timing of the exile of the Red Sitters. We see from Toveine's POV in TPOD that she recalls her exile on a farm as lasting "twenty years," [TPOD: 26, The Extra Bit, 513] but that doesn't wash with the rest of the continuity, since we know that Owyn at least was gentled only fifteen years ago [TSR: 17, Deceptions, 195]. If Toveine and her fellow Sitters were exiled twenty years ago, that places their exile immediately after the War and probably even before Moiraine and Siuan learned of the BA murder campaign. This doesn't make sense, because if the murder campaign was the reason for their exile, it seems difficult to believe that they would have gotten off with mere exile. Furthermore, in ACOS, Elaida mentions that all three Sitters went into exile 15 years ago, which fits more squarely into the timeline [ACOS: Prologue, Lightnings, 21]. It's likely that Toveine was just exaggerating or rounding off to the nearest decade (or RJ slipped up).