So far, my Paladin is winning.
Not necessarily because she has the most sets at least partially completed already — I haven’t bothered to tally that up — but because she was the one whose sets I have felt the most motivated to get out and try to “complete” when I looked at them. I put “complete” in quotes because I consider that I have enough of a set when I have shoulders, chestpiece, belt, leggings, gloves, and boots. I don’t care about bracers because they usually aren’t seen, and I usually don’t care about hats because I like to see my characters’s faces.
Sets that my Paladin already has “complete” to my satisfaction include Molten Core, Sunwell, Trial of the Crusader, Heroic Blackwing Descent/Bastion of Twilight, LFR Siege of Orgrimmar, and Normal Highmaul/Blackrock Foundry (yay follower missions!).
During this week, my last week of full leave from work, my Paladin has run Blackwing Lair, Battle for Mount Hyjal, Black Temple, Heroic Firelands, Temple of Anh’Qiraj, and Karazhan. She also popped into Normal Dragon Soul and biffed Morchok to get the boots for that set, which (aside from the bracers and hat) were the only piece she was missing. Her Firelands run netted her the last couple of pieces for that set. When the sets from the above mentioned raids are “complete”, then she can work on finishing the Naxxramas 25 and LFR Heart of Fear/Terrace of Endless Spring sets, and she can start working on the Serpentshrine Cavern/Tempest Keep set and maybe the Normal Heart of Fear/Terrace of Endless Spring set. For all the other sets, I’m pretty much happy with just having the shoulders and using them to play mix-and-match with other items.
When my Paladin is done, I think my Druid will probably be up next.