We have 13 Skulls confirmed so far. (In alphabetical order only)

Black Eye: Shield doesn’t recharge until you melee enemies
Blind: HUD and first-person arms/weapon hidden
Catch: Enemies are grenade happy
Cowbell: Acceleration from explosions increased
Fog: Motion sensor is disabled
Famine: Weapons drop less ammo
Grunt Birthday Party: Head shot on a grunt
Iron: Cannot revive/be revived (No word if you die in solo play will it revert to beginning of mission)
IWHBYD: Rare dialog is more common
Mythic: Increased enemy health
Thunderstorm: Enemy abilities upgraded
Tilt: Enemy resistances/weaknesses enhanced
Tough Luck: Enemies berserk, always dive, and never flee