I keep having this idea and forgetting it. I've been ignoring souls lore mostly pending nightreign, which I will consider canon, same as the comic books and slashy Souls. But consider it an outline and happy Easter.
Petrification showed up in ds2, and figured pretty heavily in the story. Straid was petrified centuries before we arrive, rosabeth was petrified (possibly) days or months before we show up.
It's instant death, mechanically, which was caused by the curse in the first game. In the second game the curse functions as the -5% total hp penalty you suffer on death. It's also an example of lore contradicting mechanics as it should be a permadeath for your PC.
I think the lizard guys in Amana are turning into basilisk, and basilisks cause petrification. The layout of the level has these guys tied to the milfanito, and the final tower with the weird statue, milfanito, and great swamp pyromancers is surrounded by basilisks.
Aldia decided petrification was going to be standard opsec for his territories (lion soldiers, ornifex, the gremlins, the giant basilisk in the keep, the eyeless basilisks in tseldora...)
The kingdom of the rat would seem to be the faction most cosely tied to this school of "magic".
As a school of magic it seems to be nestled between stone/earth magic and death magic, it's associated with "vermin" in its natural state (rats, basislisks), but is also associated with some of the smartest royal-court-associated characters. It seems to be opposed/countered by "life" magic, but in particular the light green plant magic we find associated with CHLORANTHY and stamina regen via the fragrant branch, rather than Izaliths various visions of fire and fire's own tendency to lead to petrification (demon ruins, ds3 asylum demon).
Right, so,
Castle Drangleic is full of soldiers that are undergoing or have undergone petrification. These include Syan's men, both the 'sunlight' heavy knights and the greatsword soldiers, and the primal knights if you want to include those (they have almost identical armor as the Syan knights, apart from the helmet), as well as Velstadt's soldiers and Wellager's men (if that's who the horse headed twinblade soldiers answer to). Soldiers that don't petrify seem to turn into ghosts, a state that seems to be described by the darkdrift katana as existing imperfectly between worlds.
This petrification maaaaaaybe relates to the Milfanito, but I think the lullaby is only serving to pacify the inhabitants, as we see in Shulva and Amana.
So the petrification may be related to Nashandra, but she seems more related to the curse of hollowing, not of petrifying.
I don't think there are any actual statues that require a fragrant branch in the Castle, which is somewhat interesting, as it seems like maybe since the soldiers in the castle are petrifying more slowly than curse-frogged - Rosabeth, Straid, the Lion Clan - that once they fully petrify there's no coming back.
Santier's Spear describes a living statue, and the 'rock' the speartip is embedded in would seem to be the head of this same stone knight. They don't seem to be a golem, so the other conclusion would seem to be some kind of living statue. Which we do see something like living statues in the first and third game, very obviously with the 'baby firesage' demons, but in Lothric petrification seems to be a fate that befalls most-if-not-all older demons when their flame goes out.