Complete the indicated task to unlock the corresponding costume.

Big Time
In Times Square, search behind the red bleachers, on the glass.

Classic Black
To the left of the main Oscorp Building downtown, look under the Gazebo in the narrow park.
Alternately, complete 50% of Campaign mode.

Classic Spider-Man
Begin playing the Rhino Challenge DLC.

Cross Species Spider-Man
Successfully complete the game.

Future Foundation
One block right of the north bridge, search a small alley behind the gas station/store.
Alternately, complete all crimes and green side missions.

Movie Vigilante
Begin playing the Lizard Rampage Pack DLC.

Negative Zone
Search the top of Beenox building, (second building in from the Brooklyn Bridge).
Alternately, complete all Xtreme challenges.

New Black Suit
Have a 100% game completion.

Scarlet Spider (2012)
Note: The camera is required. In Central Park, move north from the fountain until you see a bridge to find the costume.
Alternately, collect all 700 comics.