Hal_Taffin

Umbral Aura: Inky Aspect
Powers\WarHold2_loop.ogg
Powers\WarshadeShield_loop.ogg

Sonic Attacks: Shriek
Powers\SonicSmallBlast3.ogg

Sonic Attacks: Scream
Powers\SonicScream1.ogg

Sonic Attacks: Howl
Powers\Sonic12.ogg
Powers\Sonic11.ogg

Sonic Attacks: Shockwave
Powers\sonicgale.ogg

Sonic Attacks: Shout
Powers\Sonic14.ogg
Powers\SonicMassiveBlast.ogg

Sonic Attacks: Sirens Song
Powers\SirenSong.ogg

Temporary Powers: Throwing Knives
Powers\knifetwirly.ogg

Fiery Melee: Fire Sword
Powers\FireSword4.ogg
Powers\FireSword_loop.ogg

Fiery Melee: Fire Sword Circle
Powers\FireSword4.ogg
Powers\Fireball1.ogg
Powers\FireSword_loop.ogg

Fiery Melee: Greater Fire Sword
Powers\firestreaks3.ogg (the high pitch part)
Powers\FireSword_loop.ogg
Powers\glow3_loop.ogg

Energy Blast: Power Bolt
Powers\PowerBolt11.ogg (activation)
Powers\explohit2.ogg (hit)

Force Field loop
Powers\ForceField3_loop.ogg

This applies to...
Force Field: Personal Force Field, Insulation Shield, Dispersion Bubble, Force Bubble
Robotics: Protector Bots' Force Shield

Energy Blast: Power Burst (activation)
Powers\Mega6.ogg

Out of Range (As noted by Shadowtech)
FrontEnd\Range1.ogg

Tarzan Emote (Also used in Unstoppable, Integration, Tough)
Female: Powers\taunt4.ogg
Male: Powers\Taunt5.ogg
Huge: Powers\taunt1.ogg

Ice Manipulation: Ice Patch
Powers\IcePatch_loop.ogg

Later in the thread, NeonMan has the best Portal Corp sounds list. (Teslaroom3_loop.ogg is not used.)

Confirming Supernumiphone's finds:

Psionic Assault: Psychic Shockwave
Player\FemTaunt2.ogg

Cold Domination: Arctic Fog loop
Powers\fog_loop.ogg

Hero Police Scanner
Weapons\PPD1_loop.ogg


And some older requests:

Plant Control: Carrion Creepers
Sewer\Tentacle7.ogg

Dark Miasma: Tenebrous Tentacles loop
Powers\tentacle_loop.ogg

Willpower: Mind Over Body
V_Powers\reconstruction.ogg
Powers\voices2.ogg
Powers\Voices.ogg
Powers\regen5b.ogg
Powers\Heart_loop.ogg
Powers\glow5_loop.ogg

Psionic Mastery: Mind Over Body
Weapons\mindoverbody.ogg

Psionic Assault: Mind Probe
V_Powers\psionicMindProbe2.ogg
Powers\infuse4.ogg (short sound used when most psi powers hit)

Yes, they cover all three pets and the 2nd upgrade. The WalkieTalkie that plays seems to be randomly picked.

Mercenaries: Summon Soldiers/Spec Ops/Commando and Tactical Upgrade
Weapons\WalkieTalkie1.ogg
Weapons\WalkieTalkie2.ogg
Weapons\WalkieTalkie3.ogg

As Nod mentions in this post, some power activations will play the hasten sound while hasten is running. Tar Patch is one of these. hasten2.ogg will cover it. hasten.ogg is not used (in hasten at least).

Dark Miasma: Tar Patch
Powers\miasma3.ogg
Powers\miasmaburst4.ogg
Powers\ForceField5_loop.ogg

Dark Miasma: Fearsome Stare
Powers\aura4.ogg
Powers\darkfear2_loop.ogg

Dark Miasma: Petrifying Gaze
Powers\petrify1.ogg
Powers\miasmaburst.ogg
There's also a quiet ambient loop that I'm not going to track down.

Radiation Emmision: Choking Cloud loop
Powers\Tar_loop.ogg
Powers\Rumble_loop.ogg

Flight and Hover loops (this is the only file needed from the old list)
Powers\Flight2_loop.ogg

Player Bases: Entrance Portal
V_Base\EntrancePortal1_loop.ogg

Player Bases: Overall Hum
Warehouse\WarehouseAmb2_loop.ogg

Player Bases (Tech): Inspiration Collector
V_Base\compAmb2_loop.ogg
V_Base\computerx7_loop.ogg
V_Base\electricityhum4_loop.ogg
Tech\zapperM_loop.ogg
And one other low level hum sound.

Thermal Radiation: Warmth
V_Powers\BloodBoil2.ogg (the high pitch noise)
V_Powers\cauterize.ogg
Powers\buildup.ogg
Powers\Ignite2.ogg
There's still a quiet rumble on activation that I haven't tracked down. I can't get it to happen all the time and I think it's negligible so I'm leaving it at that.

Dark Miasma: Twilight Grasp
Powers\miasma3.ogg
Powers\Heal1.ogg

Robotics: Pulse Rifle Blast (this is the weaker attack with a really high pitch)
Weapons\PulseRifle2.ogg

Alarm Sound in city zone police stations
V_PPD\Cellblock_loop.ogg (peregrine island)
V_PPD\Cellblock2_loop.ogg (steel canyon)

Traps: Force Field Generator expiration
V_Explo\metal_explo2.ogg (the major explosion)
V_Explo\Boom_explo2.ogg
V_Explo\explomed2.ogg
Weapons\grexplo2.ogg
All that's left is a little bit of quiet fire noise.