For me they are in no real order.

Shotgun in Bulletstorm. Just melts enemies and a nice big boom

Combat Shotgun in Rage 2. Not a great game but the shooting is excellent. Combat shotgun has a nice weight and feel to it.

Wolfenstien the New Order. It's dual-wielding semi auto shotguns!

Turok on N64. It seems quaint now but to me at the time it felt like the only game that rivaled Doom 2 shotgun. Enemies go flying and you see the stream of blood.

Doom Eternal Super Shotgun. Powerful as all hell..erm more powerful than hell. The meat hook for grappling enemies and you can chain it.

Funny how 3 I mentioned under the same publisher but those guys know gunplay well.

Gnasher, Gears of War

Resident Evil 4

Bioshock

I second the Super Shotgun from Doom Eternal and would add that it kicks arse in Doom 2016 also, same for Wolfenstein The New Order and The New Colossus.

The gnasher from Gears of War. One of the most satisfying guns I've ever used. My favorite easily.
Doom eternal Super shotgun just wrecks everything.
RE7 shotgun was so nice to use and always my go to in the game.
Halo: CE M90 is a classic. Unfortunately the rest of the shotguns in future iterations were never as good or fun to use.

Also Doom 2 and Doom 64.

Was going to say.. still one of the best/most satisfying shotguns any game.

And most iconic in FPS.

I don't play too many shooters. Besides the Super shotgun of DOOM/DOOM Eternal, I have very fond memories of the Masterton M-557 of Goldeneye 007 for the Wii. It was a blast (no pun intended) to use: the last unlockable weapon and the only automatic shotgun in the game. It felt amazing to just charge at people online shooting multiple powerful blasts and destroying everyone.

I feel like this should really just be a list of various Doom games.  We can play around with the order.  But, when it comes to shotguns in video games, doesn't everyone immediately think Doom?

Doom Eternal has the best one because of the meat hook.

As a former Destiny player, I always liked the look and performance of The Fourth Horseman exotic shotgun. It made its return in Destiny 2. Just having 4 barrels is insane and the gun fires in full auto. It's exotic perk allows it to have increased damage and spread with each shot. It also had a catalyst that allowed it to shoot a 5th shot as well as dramatically increase the reload speed. You can see in the video below it melting enemies! 

Doom 2 Double Shotty
Turok Single Shotty
Are some of my favourites