At a glance
Tank wise blood DK seem on top with their massive self healing solid tanking shenanigans. All the tanks seem at least from my perspective when I am healing to be pretty good ATM. Some are a bit more spikey than others but all have their upsides. Blood DK are just in a really good spot at the moment and are capable of doing some things no other tanks can currently.
DPS wise Hunters seem really good if a bit boring rotationally from what I hear. From doing pug LFR and normal flex pugging even crappy marksman hunters tend to float to the top of the DPS list. DPS monks seem really high as well although they have caught a few nerfs of late.
The underperformers currently are mages/warlocks/shaman and rogues. I once in a while see a rogue pull some huge DPS out of their rears but most of the ones I have seen have not been in a good place yet. Due to how well they scale I expect that to change soon enough as the expansion progresses and that likely applies to the other under performers as well. They are tending to be the ones that scale well once they get gear with the right stats which is hard to come by at the moment.
The healers all seem to be in a reasonably good spot. I have not bumped into any healer type who was not capable of doing the job. Druids are their usual great healers although for the moment they are one of the healer types that can drive them selves OOM if you are not paying attention since they took away innervate. From what I hear Disc priests are in a super good spot at the moment and all seem really happy with their performance.
Tanks: Agreed. It should be noted that well played warriors can completely negate a few dangerous mechanics in the current raid, due to their incredibly mobility. Also, Paladins are the preferred tank on the instance's "Patchwerk" DPS check encounter, as they will deal far more damage than any other tank in that exact situation.
DPS wise, only warlocks are really struggling. Shamans are mediocre in general but do 20% more DPS than everyone else on a couple of key encounters due to bonkers AoE damage. Mages and rogues were weak at release, but have been buffed since and are now (along with warriors) the highest DPS classes overall if you are willing to change specs between encounters.
Healing, everything works. It's a
reallywell designed expansion for PvE healing, both in terms of play style and balance. Disc priests are kings since PWS is absurdly overtuned, but stack really poorly for the same reason, so it's not really a big problem. It's not like anyone will beat mythic with all disc priests yet.
For PvP healing, everything is fucked up with Druid being so much better than every other healer that it's not even funny. Monk healers are pretty bad in general, but can completely negate burst on a target once every 55 seconds, which makes it auto win against cool down based burst comps. Priest healers (Disc) were overpowered in PvP for a week at launch due to bugs and low gear levels, so Blizzard fixed the bugs and threw some nerfs on top, to the point where there are now something like 2 priest healers in the top 200 players.