Gunner, Warrior, Thief and Songstress are obtained at the beginning of the game, so I'll cover the rest.
Chapter 1:
1. Festavalist: Talk to Brother before going to Mt. Gagazet at the start of Chapter 1. You can also get this in the Monster Arena as a reward for the third Tournament.
2. Psychic: Obtained as a reward for being the first Monster Arena Tournament (note that you'll only ever get one).
3. Black Mage: Complete the first mission at Mt. Gagazet at it's peak; you can't miss this.
4. White Mage: Complete the mission at Besaid Island; this one's unmissable too.
5. Gun Mage: Do the mission at the Moonflow. This one is missable.
6. Full Throttle: Keep talking to Trommel in at Macalania Woods, at the spring adjacent to the Travel Agency, until he hands it over. This is another missable dressphere.
Chapter 2:
7. Alchemist: Clear the Chocobo Ranch for Clasko of fiends. You must've taken Clasko onboard the Celsius, but once he bails, this mission can be held off until endgame. You can still miss this one though.
8. Machina Maw and Floral Fallal : You'll get these in required missions; it's impossible to miss them.
9. Dark Knight: Found in the Bevelle Underground. There's a puzzle involving the lifts in the area right after the Gaol room. It's hard to describe, but I'll do my best. Ride the left lift up, drop down back to the lifts, ride the center lift, activate the switch, ride back down, take the right lift up, take the left lift down, take the center lift up and finally activate the switch. This should make the Dark Knight Dressphere available.
Chapter 3:
10. Berserker: Complete the mission at Macalania Woods. This is missable.
11. Lady Luck: Beat Shinra at Sphere Break. If you're unable to do so in this chapter, you can do it in Chapter 5.
12: Trainer: Keep talking to Kimarhi in Chapter 2 until you get the option to select something to say to him. Select the second one and you'll get this at the start of the mission here. Otherwise, you'll miss this.
13: Samurai: On the floor in Killika Temple. If for some reason you miss it, you can go back and get it in Chapter 5.
Chapter 5:
14: Mascot: Get a Chapter Complete in every area (do not do Zanarkand Ruins for last). Elsewise, you'll also get this the first time you clear the Youth League Tournament in the Monster Arena.
Once you obtain the last Dressphere you need, the trophy should pop.