Funny thing about Siren I don't think comes up much elsewhere: gaining to the top of your deck is different from putting something on top of your deck at the moment you gain it. And both are also different from something that first gains a card, then moves it (like Replace).
Topdecking as with Travelling Fair, Bauble, Tracker, Seal, Royal Seal, Insignia ::: Siren is gained to your discard, then optionally topdecked, so per the stop-moving rule it will fail to trash itself on-gain if it was moved.
Gaining to the top of your deck as with Develop, Armory ::: Siren is not moved after gaining, and so its on-gain is able to trash itself.
Gaining then after gain moving as with Replace ::: Siren's on-gain triggers before Replace proceeds to try and move it, so its on-gain trashing will happen.