Okay, first of Hello!
I've been here quite frequently as a silent observer, but now I've decided to share some of my experience, because Mystical Metal Fragments have become sort of an obsession for me. Following I'll list all of the findings of Mystycal Metal Fragment (MMF from now on) and events I can remember.
Also note that I have found most of them in free time quest.
1. Rabbit hole (hunting in forest, choosing the reach hand in hole option) couple of times MMF mostly rabbit meat, rarely chinchilla bite.
2. Squirrel leading a way (hunting/patrolling forest) also couple of times MMF, usually mithral or sometimes mana stones.
3. Travelling merchant (hunting/patrolling, either option to help or ignore) I can't remember for sure but I think there was one time of MMF, usually mithral.
4. Helping a child (patrol road to/go to shoppe, trying to reach for doll in sewer option) have gotten the event 2 times, both have yielded MMF.
5. Finding a treasure chest(patrol high seas/patrol route to, choosing the claim all treasure option) has given MMF 2 times and once some other treasure.
These are ones I can remember, I'll be sure to give an update if one arises.
Thus I have noted the following: there's a higher chance to get a MMF if your character is high in level and very focused. Also if your kingdom has developed a lot.
These were just mere theories and not in any way confirmed.
Edit: K, the developed kingdom part doesn't seem to play a role, but a higher hero level still seems to affect the chance of MMF from successful random events and succeeding seems to be affected by focus level. Still more confirmation needed though.