In my experience, it really depends on your level bracket. Early on, half-decent healers are enough to get you by and tanks are in really high demand; being a good tank and being known in your Guild as such is a great way to guarantee a spot on parties at any level though. Later on, there are loads of powerful tanks, but you might need connections to coax them onto your parties.
Clerics and other healers usually don't need the best gear to perform well in a dungeon, but they need to be able to think fast (and not lag). I'm not qualified to measure intelligence overall, but I've seen some pretty horrible healers even if the class is capped to the person's character level even above 60-65.
Among the dps classes, Illusionist stands out as a standard in serious parties (I hear Bard is the counterpart in the Taiwan server, but more on Bard later). Being able to provide MP and Atk Speed/Cast Speed support is superb and makes everybody on the party perform better (in theory). In addition, Illusionist is good at helping the AoE dps clear mobs by performing an AoE KD that disables mobs and makes them take 50% more damage now and then. Their single target dps is high.
Among physical dps, Martial Artist and Samurai stand out as the top single target and AoE damagers respectively. Again, Bard sneaks in to turn the tables if there is a Blade Dancer around, augmenting Blade Dancer's skills to compete with and maybe surpass Samurai's damage (depends on the individual). For Ranged dps, Ranger is probably the best, but any Melee dps class can don a Bow/Gun and hit from afar using high Atk Speed (perhaps boosted further by Illusioinist).
For niche specific situations, classes that are called for include Bard, Ranger, and Templar. Bard not only provides pretty powerful buffs but has a buff that prevents allies from falling asleep, and helps against certain bosses in 60-65 Trials (Raids).
Ranger is a great class if used properly and is particularly useful in Branda Trial. Not only do both bosses have a BP against Pierce, but mobs in Branda have effects that harm players who stand near them or near each other. Ranger can easily stand from far away and AoE, or use both the slow effects from Precise Shot and the Eagle summon to slow the large mobs that help to be kited. In addition, Ranger can perform powerful single target dps on the first boss while using its AoE to insta-KO the monsters summoned by said boss that kamikaze for 5000 damage on a teammate. Lastly, at the end of Branda, two Demon bugs need to be lured near crystals to lose their invincibility buffs; Ranger's Concussive Shot can push them into the sweet spot that wipes their buff quickly instead of waiting for the tank to aggo and pull them to the crystal where they sometimes stop next to the crystal without getting in range of the sweet spot.
Templar is just a nice fusion of physical and magic dps, and I personally find it useful for the end of Crimson Trial where there are some mobs that can't be hurt by magic, and some that can't be harmed physically. With Templar, I can aggro both groups long enough for my allies to easily kill them (or if the team is uncoordinated, I can handle them by myself).