Without question harmonious is the most capable in terms of soloing.

In terms of who brings what to groups, it depends on the group but typically the harmonious debuffs are pretty useful.  For solo you want (imo) the best windblade you can lay your hands on.  Ramped up parry = quivering palm on 24/7 which mean you take half damage.  Its a must.

All three do pretty good damage.  Dragon is abysmal in terms of soloability.  Drunk is a step between.

To be frank if you wanted to play a class that solo's well you'd do better sticking with your necro.  You can build them to be reasonably durable and solo some amazing things if played well and you can still kill pretty fast.

Shaman, disciple, cleric, bloodmage can also solo some pretty impressive stuff.

People typically roll a monk to pummel the crap out of things, be viable off tanks, and the safest pullers a group/raid could ask for.