(This does not award a quest point.)
Find yourself a mining pick.
A pick of some sort.
Misty Mtns: Safe in the Mine by killing A miserable orc slave (This is where most people get a pick from ).
Misty Mtns: Safe in the Mine if you don't want to get the quest point (this one requires light, but no combat).
Hollin: Helping Wagon Driver by killing A goblin soldier.
Harondor: Panther Boots from just inside the quest.
Belegost: Icholin's Tablet from close to Dimian, the young weather worn traveler.
Many quests require a pick of some sort in order to complete them, here are a list of other places and quests where you can obtain a pick.
If any of the information here is missing, or wrong, please let us know.