There are a total of 13 Suits (Costumes) in The Amazing Spider-Man 2. Unlocking all of them will earn you the Fashionista Trophy or Achievement.

Here is a list of all Suits and how they can be unlocked:

#1 - Vigilante: Available from the Beginning - 0:05
#2 - Amazing Spider-Man 2012: Available from Beginning - 0:40
#3 - Amazing Spider-Man 2014: Available from Beginning - 1:12
#4 - Superior Spider-Man: Beat all Combat Challenges - 1:36
#5 - Ultimate Comics Spider-Man: Obtained automatically during Story - 2:05
#6 - Big Time: Clear Russian Hideouts - 2:39
#7 - Scarlet Spider: Clear Russian Hideouts - 3:09
#8 - Spider-Man 2099: Clear Russian Hideouts - 3:40
#9 - Flipside: Clear Russian Hideouts - 4:18
#10 - Spider-Armor: Clear Russian Hideouts - 4:50
#11 - Hornet: Complete All 15 Races - 5:24
#12 - Ricochet: Obtained automatically during Story - 6:03
#13 - Spider-Carnage: Beat the Story - 6:35

Bonus Suit:
#14 - Electro-Proof Suit: PlayStation Store DLC - 7:10

*Depending on where you ordered the game, you may have other DLC Suits available.