Let's say a fighter with a magic sword gets turned to stone, along with all of his possessions.
Then someone chips away the stone sword from the rest of the fighter statue.
Can Stone to Flesh now be used to return the sword to, well, steel?
And on a related note:
Did the sword still radiate magic in petrified form?
Can anything be petrified in this way, even artifacts?
Flesh to stone would turn the item back to a usable item (and possibly a severed hand holding it), depending on the circumstances there might also be a maimed creature with a grudge.
The statue does not radiate magic.
Artifacts are not turned to stone, unless it would be more interesting for the story or it is directly tied to the petrified creature.
Technically, if you're only restoring the sword, you turn it into a sword-shaped lump of flesh. The former sword would be a piece of stone, which is still a valid target for the spell. However, it's not a petrified creature, so it would turn to flesh. You'd need to unpetrify the fighter to get the metal-ness sword back.
But on a level of "would this be fun/make sense to play", allowing the sword to be selectively turned back to metal would be 100% fine.
Regular stone is nonmagical, and Flesh to Stone is an instantaneous (not continuous) effect. I'd say that it doesn't radiate magic, but if you think that it should, then it can. Just keep your ruling internally consistent.
In the artifact rules, it is stated that artifacts are generally very difficult to destroy and usually require specific circumstances to do so. To uphold this generality, it would make sense if artifacts could not be petrified this way. However, the mere existence of an artifact in a campaign is an act of GM fiat, as artifacts are not part of regular loot or random loot tables. If it would make sense for the artifact to turn to stone, then it would be reasonable as a GM to do so. If it would not make sense, such as with a Sphere of Annihilation or a Philosopher's Stone (already a rock), then it shouldn't.
In my Golarion, Flesh to Stone turns the creature itself into a statue, and all it's possessions are covered in a thin layer of stone, bonded to the statue.
Magic items so encased still radiate magic, pieces broken off of the statue and subject to Stone to Flesh return to the material they previously had been.
So, yeah, in my games, chipping the magic sword off the statue and breaking the stone off of said sword would work. Same for artifacts.
Stone to Flesh would simply turn it to flesh, but Break Enchantment would turn it back into a sword.
Stone to Flesh:
This spell restores a petrified creature to its normal state, restoring life and goods...
Despite the name, StF works on items when reversing Flesh to Stone.
I'd interpret it that if you cast the spell on any part of the petrified person and their petrified gear, the whole spell would be reversed, even if the pieces were scattered.