It seems to me that a big part of the answer to this mystery hinges on chronology.

Who knew what when?

Who was where when?

First, some background:

-Lanfear devises a plan to capture Rand with a small linked circle of other Forsaken by luring him into attacking Sammael. This plan is proposed and ( more or less ) agreed to in the Prologue to FoH.

But, how was it supposed to work?

Enough clues were to be planted and enough information was to be leaked that even a simple farmboy from the sticks would be sure to realize that Sammael had taken over Illian.

Okay, so Rand attacks, then what? Things are likely to get really messy once that happens.

It seems obvious that Sam didn't go to all the trouble of building his powerbase someplace just to have Rand blow that place up around his ears.

First Premise: The actual battle was never meant to take place within Illian. Once Rand had attacked, Sammael would lead him straight into the prearranged trap, which was to be within Shadar Logoth - a place that none of the Forsaken would care about getting damaged.

Second Premise: Nobody knew exactly when Rand would attack. All any of them knew was that he was following the clues and making preparations for the attack. One of them, probably Sam, planted Melindhra and her dagger on Rand's inner circle to make sure that he didn't get sidetracked and do something else.

Third Premise: With no definitely known time for the attack, they mutually agreed on an alarm system that would notify each of them when Rand moved. They would then individually gate to a preplanned location within Shadar Logoth, link, and await Sammael and Rand's arrivals.

Fourth Premise: With the major details arranged and a Rally Point agreed to, each of them returned to their normal activities to await the tripping of the alarm.

The implication of this point is that nobody was sitting around holding hands on the day Rand attacked Caemlyn. Rahvin was in Caemlyn preparing for both attack and betrayal. Graendal was in Arad Domon. Sammael was building up his defenses along the borders of Illian. Lanfear was operating independently as usual.

The day before all of this, word gets to Rand that he thinks means Queen Morgase has been killed by Rahvin. He impulsively decides to take revenge against Rahvin before moving against Sam, and tells his key people. Mat lets slip to Melindhra that Rand is moving on Caemlyn the next morning, and she tries to kill him in order to force Rand to follow through on the attack on Sam first.

That doesn't work. She dies before getting a chance to tell anyone what she's learned.

So how does Moggy know?

Fifth Premise: Moggy spies on people's dreams and rummages through their memories while they sleep. Or, possibly she knows a technique to induce them to dream about what happened during the day. Either way, she learns about the change of plan.

Moiraine drags Rand to the docks because this day has been foretold in her trip through the Rings in Rhuidean or possibly is part of one possible future she foresaw during her Testing for AS, years ago. Lanfear is quizzing Kadere, learns about Aviendha and Egwene and their relationship with Rand, goes balistic, and she and Moiraine go through the twisted doorway together. Big pyrotechnics. No more doorway.

Afterwards, Rand attacks Caemlyn, trips one of Rahvin's Wards, and Avi, Mat and Asmo die in the resulting lightning storm. Rand BF's Rahvin back to who-knows-when, and all are returned to life.

A short time later, while the Aiel are still mopping up Shadowspawn, Davram Bashere arrives in the throneroom and meets with Rand.

Shortly thereafter, Asmo goes wandering off alone, pulls open a small door, sees someone he recognizes, and dies suddenly and mysteriously.

So, where was everybody at the time he died?

As far as the books, themselves, tell us:

Bel'al and Rahvin have been Balefired and are no longer in existence in any form;

Aginor, Balthamel, and Ishamael/Ba'alzamon have all been killed by less drastic means and are awaiting Transmigration ( although nobody but the author is aware of that at this time. );

Moiraine and Lanfear are somewhere inside Finnland - status and condition unknown;

Sammael is fortifying his positions in Illian;

Graendal is at home in Arad Domon, plotting mischief and playing with her living toys;

Moghedien, herself is a forkroot drugged captive of Nynaeve;

None of the other Forsaken have any known location at the time;

When last seen, Rand is in the throneroom meeting with Davram Bashere. Enaila and Somara are both present;

Mat and Aviendha are talking quietly in the garden that Asmo just left;

None of these five have any known motive;

All others are totally unknown.

So, what has RJ told us about this death so far in interviews, booksignings and the like?

First, that the perp should "be intuitively obvious to the most casual observer." Also, that all of the clues to the killer's identity are present before "The Fires of Heaven" ends. That the killer is somebody we've already seen, ie. met as themselves. That both where Asmo died and how he died are the reason he cannot be transmigrated. That it was not Padan Fain. That the DO didn't order an assassination.

What additional clues has he published in subsequent books?

That Demandred did not know that Asmo was dead until the DO told him. That Graendal had been expecting a visit from Moggy. That Sammael would gladly kill Asmo if only he knew where Asmo was, and truly doesn't believe he is already dead. That Lanfear/Cyndane had been "held" by the Finns.

Who does that leave from those whom we've already met when Asmo dies?

Moiraine whose possible motive in light of her letter to Rand is unknown.

Bashere whose possible connection to Asmodean is unknown.

The DO itself. While we technically haven't yet met Shaidar Haran AS Shaidar Haran, we did meet an earlier unnamed version of him in "The Great Hunt", and the new, improved, full-blown version is what greets Demandred at Shayol Ghul the very next day. RJ has also said that, while not a full blown avatar, SH *is* a vessel into which the DO can project a shadow of itself. Therefore, technically, if SH did the deed, the DO did it itself, which isn't the same as ordering it done. But, would Asmo have recognized Shaidar Haran? Unknown and unknowable. Graedal didn't, and we have a confirmed visit by her to SG, so SH must not always be present at such meetings. Semirhage and Mesaana are both unpleasantly surprised by him in subsequent books. It seems most likely that SH would have been unknown to Asmo.

That's as far as I can go. There seems to be only two real suspects and no way to pick between them other than which feels most intuitive to you.