Depends on the bracket. For 39. I have done pally/priest/druid. Shaman are a rarity, not because they suck tho, but i have little experience so they will not be included, and ive never seen any that are anything above average.
Priest: Best fast heals and emergency heal via shield and renew never sux. They are also the ONLY healer who can remove magic, for cc/fear/dots etc. Pallies do not have magic cleanse at 39, this is huge versus cc classes and warlocks. And mana burn is just evil. You cannot however remove many melee slows and dont have the armor to do too well versus stun lock. You are also forced to spec for the regen while casting in the disc tree or you will just go oom super fast which means you lose out on many of the heal increasing talents in holy. This also means you need alot of stats to do well as a priest, you need spirit and intellect both in good amounts to keep mana up, you of course need +spell power, you also need good stamina since you dont have much armor and just eat the stun locks. My lowest rated of the three healers, dispell alone tho is so killer on top of mana burns to stop other healers makes them still very good. You cant however remove poisons like mind numb, wound, viper sting, crippling, etc nor can you remove curse of tongues. So you can be more easily mana drained or have your heals take forever to cast or have healign debuffs you cant remove put on you or those you are healing, pallies and druids can remove most of if not all of these.
Druid: Great heals, HoT's and jsut about the best survivability due to the fact that you can just shift outta any slows into travel and get away from anyone except ret pallies with justice, no reason to die to anyone really unless you have justice on you. On top of that you get an instant heal with your HT for a good 1500-2200 heal every 3 mins for emergencies. Your regen while casting talent is in your main healing tree so unlike the priest you give up nothing for it. The spec can also do great damage if you can spare 5 pts, a simple spec into the 5 to reduce wrath cast time and your healer is now one of the best dps'ers out there, no joke. You can also use cat form to hide when at a flag to avoid being sapped or jumped first because you are a healer. And when geared and spec'd right for int/spirit, your regen while casting talent will let you heal for a very long time before going oom. Its also one of the most fun healers with so many options and so much involved. And you can remove curses and poisons, helping remove some rogue and hunter poisons and warlock debuffs (slows/mana burners/cast time increasers/damage/wound poison).
Paladin: Pallies are a fairly boring healer, sorry. But they work very well when you dont be an idiot and gear with cloth and leather like many Loladins do. Your strenght is your cheap heals and good armor and 100% pushback resistance. You will rarely cast holy light, keep that in mind. You will be using flash spam 99% of the time. And sadly, you have no other spells other than purify to remove poison or disease. You can remove rogue nastiness and hunter poisons, but again you do not get magic cleanse at 39 so warlocks will eat you alive with ease. Your gear focus will be mainly to keep decent hp in the just under 3k ish area, and decent mana, but you dont need to go overboard as your heals cost very little. Ignore spirit, you almost never get ot use it. Much more than other healers you need +spell power to get your flash of light to do enough to matter. Even things like the +16 weap oil (+30healing equiv) is a necessity as is the seal of light glyph for another 5% healing. Dont forget your +5% crit on flash glyph either. You should be able to get near 450-500 per flash with good gear. While still keeping most everything mail armor. The only gear that you could sub out would be the engineering goggles and some revelosh leather shoulders. I dont even use the revelosh gloves as its just too much armor and intellect loss when combined with the other losses. Your strength is its hard to go oom unless its just a zillion year fight as long as you just spam flash. You dont get pushback so the heals are on time, and your armor is near 3k so you have good mitigation to survive stun locks and focus fire. The only way you can get takin out is with some cast time increaser debuffs or some MS that is accompanied by lots of twink dps on you. With good teamates to take out those on you, you will live long enough even with the debuffs to get those guys takin out before you. What you give up in utility and mobility (and dps, you are the worst dps class and spec by far with holy after the SoR nerf), you make up for with mana that just doesnt run out, and survivability. It is the easiest healer to play too, bar none.
Overall, the druid and pally are the best, each has its ups and downs. Because druid dps is the highest of any class (but you go oom nutty fast), I have redone my druid to balance and use my pally for heals. But both works just fine. The priest could be good, but needs a bit more support due to the lower survivability under focus and her weaker heals due to forced talent pts into disc instead of mainly holy. But her dispell and other utility do help to compensate.