Flak cannon for the win. Unreal Tournament (UT99), while being my first FPS, is also one i loved. When i see PvP in recent FPS, i feel like we lost so much. Obviously they're not in the same subgenre, but still. I've an issue with unbalanced PvP. And nowadays, most PvP games are just that.
While i enjoyed every hour passed in the Unreal franchise: I also played the solo experience Unreal is, or Unreal Tournament 2004 (UT2k4) with some Assault, i've increasingly moved away from PvP games as a whole. I still do remember the fun we had in computer rooms playing Quake III Arena together. Slightly more extreme than UT99 but i didn't play it enough to savor it fully, even if i experienced the techniques at least once like the plasma climb and the crazy speed obtainable with bunny hopping.
Now, the game i played the most in the shooter genre is a hybrid of TPS and Action-RPG: Warframe. And funnily enough, the developers, Digital Extremes also co-developed UT99. While the farming aspect got me for an unreasonable amount of time, the game has its undeniable qualities. Game feel / movement techniques in particular. In a similar fashion of what Grinding Gear Games do with Path of Exile, new content keep coming periodically. GaaS are more and more common after all.
Second to last in the list will be Control. Brutalism is an architectural style i always liked. And it slightly influenced how the site is. The atmosphere is probably my favorite aspect of the game, but the gameplay isn't left behind, and once you get most abilities it feels really satisfying.
To conclude, i'll name a game and more specifically a mode of it: ShootMania Obstacle. It's a racing parkour FPS game mode. A parallel can be made with TrackMania regarding the racing part only. Racing part which can be facultative as it's both hard to learn and to master the game. There's no better feeling than improving your skill set there, and the skill ceiling is insanely high. If you're curious, here's a run done by SBVille under 10 minutes on the map Faith Improvisus.
Note: The listed games are those which spontaneously come to mind. The purpose is to give you an insight of what i like the most in this genre.
Credits: Unreal Tournament by Digital Extremes and Epic Games, Warframe by Digital Extremes.