What is Karthus's Phylactery?
For those who do not know, the Phylactery is an object that a Lich stores their soul in so that they can return from the dead. So, what does my main man 
use? Is it his book? His staff? Will we ever know?
11 Comments
Dont apply lore of other games, books, religions or legends to the lore of LOL. That thing does not exist in Runeterra as far as we know.
Karthus is a lich only to fill that niche in game
In the lore Hecarim, Kalista, Thresh and the rest of The Black Mist undead function exactly the same. The Mist is what keeps them in a state of unlife.
This is a question better suited to Mordekaiser because he isn't a Black Mist undead spirit instead he defied death by his own will.
Karthus very much enjoys being dead as that is pretty much all he really admired. His spirit form granted him just that and he does not want to be alive ever again so storing souls and returning from the dead like Morde would want is not for Karthus.
It would be the Black Mist. Even though the Black Mist is by no definition a phylactery, it would be the closest 'equivalent' to OP's suggestion.
The Black Mist is the source of all the spirits and wraiths of the Shadow Isles. When they're defeated, expelled or whatever the spirit eventually reforms within the Mist. The stronger the willpower or malice of the spirit, presumably the quicker it ends up coming back.