This list is subjective of course, feel free to add to or question it. Just my opinions and observations.
Hunter - S-tier
Available to every race in the game. Able to use every weapon type except maces and wands. One of the most popular classes- almost always in the top 3 played, at least. Pets classes are something most people seem to enjoy, and Hunters currently have access to 1,932 different pet appearances (and counting) at the moment. They are also the only class to use Guns, Bows, and Crossbows, which lends to further uniqueness. A few legendary weapons here and there. No weapon illusions, however.
All Plate Classes: Paladin, Warrior, Death Knight - A-tier
Available to every race except for Paladin, and they all have very solid, if not the best tier sets consistently. Also being able to use most Legendary-quality weapons, from the earlier expansions, 2-handers are some of the best cosmetics WoW has to offer. Each class can use weapon illusions, with DKs having some of their own unique ones, and Paladins and Warriors can use Shields. DKs also have further customization options with various minions and undead they can alter through glyphs and talents.
Shaman, Monk, Warlock, Druid - B-tier
Warlock and Monk available to all races, with Druid and Shaman being a lot more specific. Warlock (pets) and Druid (forms) have access to unique barber shop customization options, with some very cool choices. Shaman and Monk have access to special weapon illusions and glyphs that chance the colors of certain attacks, and Shamans’ Ghost Wolf. Warlock also has the green fire questline. There are a handful of Legendries than can be used by these classes. Each of these classes also has some very distinct looking tier sets. Monks can also hide their weapons completely, and use their fists. The only downside, is that Druids will likely be seeing their bestial forms a lot more often than their humanoid ones.
Priest, Mage - C-tier
They are available for every race, but fairly standard when it comes to customization. Some very solid tier sets. Some decent glyphs, and books for Mages that change the appearance of the Polymorph spell. The lack of cool transformations and pet customization puts them a step below the other spellcasters, however. They can use just about as many legendries as the B-tiers. Robes are limiting, though.
Demon Hunter, Evoker - D-tier
DHs can only be elves, and specifically Nelves & Belves. Evokers are locked to their own new, specific race (Dracthyr), which is basically a Blood Elf/Dragon hybrid. While Evokers and their visage forms easily have the most customization when it comes to race, this is countered by the fact they can only display their shoulders and belt, and cannot wear helmets nor anything else. DHs do have access to Warglaives as a transmog, and Evokers have a legendary of their own. Both have some very intense spell effects. They would be higher, but are held back by the races that can play them.
Rogue - F-tier
Available to every race, as of recently. Rarely ever in WoW has a class actually lost customization options, but here we are lol. Back in the day, in the three earliest versions of WoW, Rogues could use ranged weapons, the same as hunters, but also could use thrown weapons- this included throwing axes and knives, shuriken/ninja stars, and the like. Now, all of that is removed, and they use a generic pirate-themed pistol, and static thrown weapons and their ranged abilities. Rogues are also very generic when it comes to how their visuals are, and unfortunately very little has changed with them over the years. I honestly think they are consistently the least popular class in WoW, in the current era. Two specs, Assassination and Subtlety are also limited to Daggers only.