Note that I'll only be talking about those classes that have been central options available in multiple editions. The class has to have been in the Player's Handbook or equivalent, and have appeared in more than one edition in such a capacity (so no cavaliers or gunslingers).
WarriorsBarbarians are completely unnecessary, and the RPG archetype is ridiculous; even Conan the Cimmerian, one of the most famous "barbarians" of fantasy literature and film, wore armor when it was appropriate. In my opinion, there's no need to have them as a distinct class from fighters.
Paladins are basically fighter/clerics, with the weapon selection of a fighter and some of the spell and turning ability of the cleric. Tonally, they might as well just be clerics, so in my view they aren't needed as a separate class.
Rangers' tracking ability and (in some editions) proficiency with two-weapon fighting are nice extra abilities, but not essential in a primarily dungeon-focused game. They might not be needed as a separate class from fighters.
PriestsDruids are a tough case, as besides their extra powers (like shape change), they sometimes have a very different set of spells from their parent class; in oD&D and AD&D 1e, they don't even get cure light wounds until second or third level! The issue is that having both clerics and druids can lead to some tonal mismatches. I might elaborate on this in a different post, but the jury's out on the necessity of druids.
RoguesAssassins are in a similar boat as rangers, as they're basically normal thieves with some extra abilities. Assassination is largely redundant considering thieves can already backstab, but their disguise ability has so much potential; admittedly, it could just as easily be assigned to regular thieves instead (shades of Lupin the 3rd?). Since they're also of little extra use outside of cities, I'll call this one a draw.
Bards, on the other hand, are excellent to have in one type of game: one with a small number of players. In a group of six, it's easy to have at least one player character to fill each basic role, with some doubling-up as needed or desired. This is harder to do in a group of three or four, and while the bard is certainly no substitute for a fighter or a spellcaster (except in AD&D 1e, but that's a whole other kettle of fish), they can certainly pull their weight in a pinch. The problem comes when bards are designed to be equal or superior fighters, thieves, or spellcasters compared to actual, single-classed fighters, thieves, and spellcasters; 5th edition has this problem in a big way. I'll chalk this one up as a maybe due to their utility for some groups.
WizardsIllusionists, if their spell lists are different enough from standard mages, can have interesting possibilities. I'm not a big fan of them in AD&D 2e due to the fact that mages can access every spell, with illusionists' (and other specialists') only advantage being more spell slots and easier learning. What are not different enough are...
Sorcerers. This step-headed redchild is so bafflingly similar to its parent class that both sorcerers and wizards share exactly the same spell list in D&D 3.5; even in 5th edition, their selection of cantrips is identical. If you prefer the sorcerer's more cleric-like method of casting spells, that's fine. But the presence of both sorcerers and normal wizards in the same setting destroys any reason to play the latter, and also destroys the entire rationale for wizards being somewhat rare and secretive.
The Monk (or Mystic)Monks are... an odd case. AD&D treated them as the fifth core class, and with good reason: their means of attack, defense, and other abilities can't be easily slotted under the warrior, priest, rogue, or wizard groups (although some editions, including oD&D, consider them a cleric sub-class for attack and hit dice purposes). Even in BECMI, the rechristened mystic was added as the only other human class available at first level (the druid, as well as the name-level fighter trifecta of paladin/knight/avenger, being more akin to WOTC's prestige classes). Tonally, they might not fit certain settings in their default form, but I would argue for their inclusion.
The Final Count-upAssassin - Maybe
Barbarian - No
Bard - Maybe
Cleric - Yes
Druid - Maybe
Fighter - Yes
Illusionist - Maybe
Mage - Yes
Monk - Yes
Paladin - No
Ranger - Maybe
Sorcerer - No
Thief - Yes
(Reminder: this is all my personal opinion. Feel free to disagree, but please do so civilly.)