Which abilities don't share cooldowns?

There's technically three types of powers in Mass Effect: Active Powers, Grenades and Passive Abilities.

Active powers a single Global Cooldown. A skill either activates the Global Cooldown or it doesn't. Skills that don't activate a cooldown are only limited by how quickly you can perform the action (the time it takes to switch ammo/throw a grenade).

Most powers listed as a base power here qualify as Active. These are your standard attacks and always cause a cooldown.

Passive powers are always active and don't have cooldowns, simple enough. Ammo powers are technically passive, and you can always switch ammo regardless of cooldown. Tech Armor is basically a passive power unless you detonate it.

Grenades don't use a cooldown either, and if something seems like an "active" power and it doesn't activate the cooldown, that's because it's technically a grenade. Proximity mines, Nova and Sentry Turrets are all technically grenades.

Nova has a "cooldown" that's actually based on your Biotic Barrier (the purple "shield" Vanguards have). You can't use Nova if your Biotic Barrier is empty, but it otherwise ignores cooldowns like a grenade.

Ammo Powers do not trigger a cooldown when activated, but cannot be activated when something else is on cooldown.

Fortification, Defense Matrix, and Tech Armor can be turned on without causing a cooldown, but activating their ability (+melee damage, shield restore, explosion) causes a cooldown.

Grenades and Nova are on their own separate cooldown class, and can be freely intermixed with powers. (Nova might require shields, though)

Sentry Turret and Combat Drone (engineer specials) share a cooldown with each other.

Finally, some highly evolved powers have a chance to proc a bonus power, where activating them will not cause a cooldown. (Examples: Tactical Cloak, Charge, etc.)