Honestly it needs to be divided because single player shooters and multiplayer have very different designs and goals, its why one tends to excel at one part and not the other, especially as you get more and more recent.
However for multiplayer specifically
God I wish someone would take anouther crack at this, the high speed mobility mixed with weapons that tend to be devastating but require skill to land hits is just fantastic and never gets old.
For single player its got to be...
Just Half-Life but better in every way, more enemies, more weapons, more interaction. If Half-Life cemented FPS as the most popular genre Opposing Force perfected it.
I may be in the minority here but i did enjoy Black back on the PS2 and original Xbox era. Fantastic game ahead of its time. Resistance and Killzone are definitely up there with Halo 3 as well.
Doom Eternal (like playing a multiplayer match of Quake - only with monsters!)
Halo CE (the only Halo that has a REALLY good single player campain imo)
Half Life 2 (still more interative than modern FPS)
For multiplayer
Quake 3
Halo 2
I am glad there is barely a mention of Destiny - those games are not really FPS. They are "looter shooters" where the actual focus is looting and the enemies are mostly just bulletspunge pinatas with goodies inside