Co-op babies in the Binding of Isaac

Listing only "special" characters; These are the ones that will show up as explicitly selectable, not the "normal" question mark cosmetic characters.

Entries with a dark background have to be unlocked in some way. Currently, you'll have to look elsewhere for unlock conditions.

Rebirth

Randomly:
Friendly spider
Death effect:
Drop heart
Drop soul heart
Always:
Pretty fly
Death effect:
Tear burst
Always:
Homing tears
Always:
Attract enemies
Attract enemy tears
Randomly:
Troll bomb
Death effect:
Troll bomb
Floor effect:
Scramble attack direction
Always:
R U A WIZARD?
Death effect:
Enemy spiders
Always:
Red creep
Death effect:
Red creep burst
No known effect
Always:
Touch damage
Always:
Belial tears
Death effect:
Drop random pickup
Death effect:
Drop coin
Always:
Tech.5
Hurt effect:
Fart
Always:
Attract coins
Death effect:
Drop bomb
Hurt effect:
Explode
Always:
Fear shot
Shot speed down
Kill effect:
Blue fly
Death effect:
Drop heart
Always:
Invulnerable
Despawn effect:
Removed from play
Floor effect:
Transform into a random normal baby
Always:
Can interact with pickups and chests
Blindfolded

Afterbirth

Spawn effect:
Bonus black heart
Spawn effect:
Bonus red heart
Spawn effect:
Bonus eternal heart
Spawn effect:
Bonus soul heart
Randomly:
Damage up (2s)
Randomly:
Tears up (2s)
Randomly:
Speed up (2s)
Randomly:
Range, shot speed, tear height up (2s)
Randomly:
Luck up +2 (2s)
Randomly:
Size down (2s)
Randomly:
Size up (2s)
Randomly:
Butt penny (2s)
Randomly:
Transcendence (2s)
Randomly:
Hive mind (2s)
Randomly:
Guillotine (2s)
Spawn effect:
Rainbow poop
Randomly:
Whore of Babylon (2s)
No known effect
Randomly:
Anemic (2s)
Death effect:
The Sun
Always:
Succubus
Randomly:
All stats up (2s)
No known effect
Randomly:
Damage up (2s)
Randomly:
Book of Belial (2s)
Randomly:
Steam Sale (2s)
Randomly:
Goat Head (2s)
Always:
Head of the Keeper

Afterbirth†

Always:
Blue fly
Always:
Pretty fly x2
Always:
Bone club
Always:
Spectral tears
Bound