Here I will show you where the best equipment for each class is, since the latest update.
All except Necromancer, Bard, Warrior, Cook
At the end of the game, after defeating all the clones, you get four keys. Head back to the apartment floor, and open the previously locked doors.
Necromancer
Livingbone Sceptre and Buckle
To get these items, you must first have the Lich Hand (undead sidequest, gold rank), and Sorcerer and/or Warrior. At the Ivory Fort secret entrance, there is an alchemic reagent gate. Go past it, and go to where the Necromancer spirit is/was. In the room, there are 5 small, outlined buttons in a row. From left to right, push them in this order: 3 1 4 2 5. After that, proceed through either the left or right path. Interact with the Lich (this is why you need the Lich Hand!) Be careful fighting him, as he is tough. After the fight, collect the sceptre and the buckle from two of the chests.
Livingbone Tiara
At the last area in the game, go to the candyland section. Outside the candy building, there is a lever hidden behind one of the trees. It will open a secret passage to the tiara.
Warrior and Bard
The Warrior and Bard's best equipment can be bought in the town south of where the Plateau Lance was found, which was recently added. The Bard's best equipment is also scattered around the game as well.
Bard
First, get a trading pass (explained above) Then, head to Marcus Fatum's tower. The left salesman sells the Ntv Awards Lute and Hat. However, I do not know where a 'Ntv Awards Songsheet' is. However, you can get the Festivalclub Score in the house that you find the Bard spirit in.
Cook
To get this guy's equipment, head to the office level of the final area. There is a cafeteria at the very left, go to it. There is a secret passage at the bottom-left section that leads to the equipment.
You can also find legendary equipment in the final dungeon, which are the best equipment in the game. It seems that you can only find 2/3 legendary equipment for each class. They are stored in brown chests, or sold in shops. These shops are in the blue truffle area behind the wealthy door, and in the giant ivory castle, hidden as only a crack behind the throne.