Cannith Crafting/BtA base items

Quest chain reward lists

 * The Lost Seekers - anything, notably one type of rare material weapon or shield seems to be always included on this list
 * Catacombs - anything
 * The Seal of Shan-To-Kor - anything, includes belt and helmet/hat
 * The Sharn Syndicate
 * Tangleroot Gorge
 * Phiarlan Carnival
 * Sentinels of Stormreach - every third reward list only
 * Delera's Tomb
 * Ruins of Threnal
 * Reign of Madness - always a ring included

Single quest/raid reward lists

 * The Chronoscope - weapons, clothing and jewelry
 * The Sharn Syndicate quests are the easiest place to get items, since each quest is rather short and easy to farm, and gives a choice of several BTA items, rather than having to go through a whole chain - at the end of which you'll likely want to get a named item. However, you won't get a belt or helmet/hat from this chain.
 * Stand Your Ground - cloaks, gloves, rings and necklaces
 * The Bookbinder Rescue - boots, bracers and goggles
 * Dirty Laundry - armor and shields
 * Repossession - armor and shields
 * The Red Fens quests - any
 * Fathom the Depths - boots, necklace, possibly any
 * The Claw of Vulkoor - helm, ring, gloves, necklace, possibly any
 * The Last Stand - shield, weapon, helm, armor, ring, necklace, goggles, possibly any
 * Into the Deep - ring, gloves, belt, helm, bracers, armor, possibly any