First, we know that BA can violate the three AS Oaths with impunity:

• They can violate the Third Oath: In [TSR: 38, 863, Hidden Faces, 438], Liandrin wishes she had the ability to kill with the Power, like Chesmal (another BA). This implies that Chesmal can, and has done so. "New Spring" makes it seem very possible that Chesmal killed a sister named Meilyn, one of Tamra Ospenya's searchers. [L:NS, 669]
• They can violate the First Oath: In [TFOH: 34, A Silver Arrow, 394], Liandrin attacks the wounded Moghedien, attempting to Compel her. Moggy strikes back, and Liandrin says, "Y-you do not understand, Great M-mistress...I only wished to help you to have the good sleep." This was definitely not her intention. Her intention was to have Moggy be her obedient slave. A few pages later, Liandrin tells Moggy that she will be Moggy's "faithful dog," and in the next sentence tries to get Temaile and Chesmal to try to betray Moggy. If she meant one, the other must be a lie. In [TFOH: 19, Memories, 260], Alviarin says to Fain, "Now answer my questions, or two corpses will be found here in the morning instead of one" (the one being the dead Accepted). Fain thinks to himself, "There would be two in any case, whether he answered her with suitable lies or not; she did not mean to let him live." Obviously Fain thinks she can lie, and do you really think Alviarin meant to let him go?
• In [ACOS: 40, Spears, 626] Galina makes it clear that the BA are not bound by the Three Oaths: "She had broken free of the Three Oaths on joining the Black Ajah, replacing them with a new trinity..." So, the BA are not bound to the Three Oaths that the non-black AS swear.

However, the BA still have the Ageless look and the shortened lifespan which are characteristic of people bound by the OR. They have to, in order to blend in with non-black AS. So, it is likely that the "new trinity" of Dark Oaths sworn by the BA are taken on the Oath Rod. There is further evidence for this. Galina knows that Oath Rods can be used to remove Oaths: "If [Sevanna's "binder"] was a second Oath Rod, it could be used to remove any oath she swore now" [TPOD: 11, Questions and an Oath, 255]. This is not general knowledge among AS (Seaine and Pevara, both high-ranking AS, had to figure that out by themselves), so we can conclude that she knows it from experience. That is, the BA are freed from the Three Oaths via the Oath Rod. It means that the BA induction involves the Tower's OR-- Galina is only familiar with the Tower's Rod (Number 3). It's therefore reasonable to guess that the Black Ajah Oaths are administered via Oath Rod.

So what are these Black Oaths?

According to Verin, one of the Oaths is "I swear not to betray the Great Lord, to keep my secrets until the hour of my death." [TGS 39: A Visit From Verin Sedai] This explains why the golden girls can't get Ispan to spill her guts about the BA no matter what they do [TPOD: 20, Into Andor, 399].

Also note Joiya Byar's "confession" about the supposed BA plan to use Taim as a False Dragon to discredit Rand. That obviously never happened, and so it appears that Joiya lied, while her fellow prisoner, Amico, told the truth about the plan to get something in Tanchico to harm Rand. This fits well with the idea that the BA swear an Oath not to betray the Ajah. Amico, having been stilled, was released from any Oaths that she took on an Oath Rod. Thus, she could spill the beans when pressed. Joiya, however, had no such release, so had no alternative than to either lie, or to grovel saying "I'd LIKE to tell you, really, but I can't!!" [Laura Parkinson]

For the rest, we can make reasonable guesses. There is almost certainly some kind of oath of servitude to the Great Lord, the same kind of oath the Forsaken have spoken of swearing. There is also likely an Oath of secrecy.

However, the Oath is probably more specific than "keep the Black Ajah a secret", since BA have told outsiders that the Black Ajah exists - Liandrin for instance [TGH: 5, The Shadow in Shienar, 66]. Adeleas said that in particular she could not force Ispan to reveal the names of other Black sisters no matter what she did. So the second Oath is likely along the lines of "do not reveal the identity of any Black sister to someone not sworn to the Dark Lord", or some such [David Chapman].