Oooh boy, here we go…
TLDR: I like monk most by a slender margin. Druid and shaman are constantly fighting for 2nd and 3rd place, and sometimes sneak first (especially back in 9.0!). Priests are a perfectly good and capable healer and I see why some people like them; I just prefer the others. Never played holy paladin yet.
If you wanted the single most support-build healer… druid or shaman. Very hard to call, it’s a choice of 2 good toolsets with little overlap.
Monk
My favourite healer for both raiding and M+. It’s mostly the pinball mobility it comes with; that rapid stop/go movement style becomes quite addictive and I strongly recommend it for anyone who often finds that the group has started moving out of a mechanic and you’re still lingering. Chi Torpedo will let you finish your cast, then move, and still be the first person out of the fire.
Outside of the mobility, it’s generally a hps healer, you pump numbers rather than mitigate issues. You do however have Ring of Peace, which solves the Necrotic affix and the angry bears in Halls. Paralyze is effective as an interrupt on 30s CD vs most dungeon trash, and will handle things like the big slimes on the first boss in PF. If you go pandaren, you can pull out a quadruple caster interrupt: leg sweep, paralyze, knockback, panda punch, and all together they really help catch trash interrupts that dps miss.
Revival is hecka strong in raids, and on Bursting weeks, cancelling high stacks across your whole group every 3 mins; with a conduit available to reduce the CD. Chi-Ji and Venthyr Fallen Order will both give you ~20 seconds of not having to pay attention to party heals and just breathe.
Our current recommended legendary is Ancient Teachings, which is the fistweaving style; but it is not mandatory. MW can gear for high mastery and be a very high HPS ranged healer with Tear of Morning instead. We do sacrifice all our DPS for this, but MW is currently last on healer DPS anyway, even if we’re going all out. I keep a mastery set on standby for spiteful/sanguine/storming/grievous weeks because it’s just easier. Ideally, go Kyrian for ranged healing as it seriously doubles down on a high mastery build and gives significant windows of absurdly strong throughput.
Raiding gameplay is probably simpler than M+ in that you’re not in a great position to be assigned special mechanics. You will be fistweaving in raids unless you want to run out of mana, so you will be in melee. For the most part, your raid leader will leave you alone and let you quietly pump HPS while only occasionally assigning you to use Revival at specific times.
Monk fistweaving features a lot of blind-fire spells. Essence Font isn’t targeted, and Renewing Mist will jump to a new target if the current one is topped off, and Ancient Teachings is splash healing. You can get through entire raid encounters having never targeted another player for anything outside dispels.
It is likely, but entirely unconfirmed, that Blizz will double down on monk as a melee healer in DF. Just deduction based on evokers being another ranged healer.
Biggest let down: PvP. Although Blizz theoretically addressed this the last couple of patches, they haven’t actually addressed it. Monks need to channel too often in PvP, and have no resistance to interrupt. Do not play for PvP. Also, Chi Bubble does almost nothing. For a 3 minute CD, you’d hope for a lot more than we get.
Druid
My second favourite. I think. Some weeks, druid feels better than monk; moonfire handles the Explosive affix very nicely. Already talked about druid utility above; knockbacks, roots, soothe, knockback 2, incapacitate; it’s good stuff 
For combat mobility, you can talent Tiger Dash on 45s CD which makes it behave a lot like monks’ Chi Torpedo.
As a ranged, mobile healer, you can pick up special mechanics in raids; although whether your RL wants healers to do these is up to them.
I do think druid is the best open world class, in any spec. Pre-flying you have stealth, and the raw mobility is amazing. It’s really hard to die as a druid, even under-geared. And if your druid is not a herbalist you’ve done it wrong.
Biggest let down: you need to watch DBM timer bars to make sure you have pre-HoTted your group before the big damage.
Shaman
Trades places with both druid and monk, if I’m honest. I’m never sure if shaman is 3rd, 2nd or 1st on my list.
As the healer with an interrupt on short CD and with good range, shaman is designed to prevent damage from happening rather than fix it later. Raw HPS might be a bit lower than monk or druid, but interrupt, spirit link, tremor totem, mana tide, purge, and bloodlust are a compelling combination. I’ve had a lot of fun with shaman in M+ since 9.0, and a lot of it is snapping interrupts on things dps have missed or ripping buffs off mobs.
Your raid lead will absolutely call for your bloodlust and spirit link, and again you’re in a decent position to soak up special mechanics, although you are less mobile than a druid.
Resto shaman is solid in the open world and probably has no need to spec dps for questing; although there’s also no reason to not go elemental since it’s the same gear. You’ll probably run dps trinkets even as resto.
I think this is my favourite healer to do random BGs with. Druid is hard to lock down, but an interrupt is a powerful thing. As are single target and AoE slows, fear breakers, purges, and ability to cast while moving. The PvP legendary is also very funny, since you can alternate instant casts of chain lightning and chain heal and the other team won’t have any options to interrupt you.
Biggest let down: if you start losing control of the situation and party health is bombing down, you don’t actually have a good strong way to fix it quickly. Shamans need to stay on top of problems and not let them develop, because if they do, then it’s on the group to help out.
Holy Priest
Currently one of the most popular healers to the point where if you have ever wanted to play this spec, I think now is the time. It’s not going to get much better than it is.
As holy, you have big hps, and with the right covenant can pull out some surprising dps as well. Many tools available, especially the tank-saving wings on a 1 minute CD, and not as slow on its feet as you’d think if you pick up Feather and have a self-target macro.
The number of buttons available can be a little overwhelming; good news, you don’t need all of them all the time. In fact there’s some that really aren’t worth the mana and shouldn’t be on your bar at all. Looking at you, Renew and Shield 
My advice is to set up mouseover macros so you can have buttons do dual duty as a damage spell to your target when you aren’t hovering over a player, or casting a heal/dispel if you are. It is the only way to fit everything on your bar.
For utility, you have a single target interrupt/stun on a long cooldown which you may never need to use. Guardian Spirit is powerful and it does feel good to catch a player who would otherwise have died. Leap of Faith and Mind Soothe are situational but strong if used well, and Dispel Magic will rip buffs off mobs. Mass Dispel cancels bursting almost as well as monks’ Revival; it has a cast time, but also a much shorter CD.
Oh, and of course, POWER INFUSION. Everyone loves you. As long as you use it on them and not yourself.
Holy Priest mastery is particularly nice and simple, just applying a HoT for a % of healing done with every spell. The spec absolutely can get by doing occasional smites, using about 4 healing buttons plus dispel and purge. Played like that it is still a strong hps healer and not too hard to handle in a group. I think you can reasonably go up to 15s without learning any of the advanced buttons.
Open world, this is probably the hardest of the healing classes. It’s hard to keep things off you and Fade does nothing without other players around - although well geared holy priests can handle ZM by virtue of big burst damage and just nuking mobs before they get near.
Biggest let down: poor self-defence. I think it is currently recommended to go either Necrolord for the shield, or Night Fae for the blink. Both are very needed, especially since the one mobility option is a talent.
Personal note: it’s not that I dislike priests… but I prefer the utility you get with druid, shaman or monk. Priests tend to look after and buff their allies (wings, leap of faith, mass dispel, PI) where others are more about messing with enemies (roots, ring of peace, interrupt, slow).
Discipline Priest
I haven’t played discipline since 8.2, and that was only in casual raids. It is fun, but it is also complicated. You need to anticipate incoming damage (DBM timer bars again) and prepare to have the healing happen at the right time.
Levelling in dungeons is challenging because it’s hard to predict damage and optimise you output when pugs are so random. There will be wipes and you will get useless comments like “spam shadowmend” and “stop trying to dps” from players who do not understand how the spec works. I actually gave up half way and finished levelling as holy to save the arguments.
M+ always felt like it was on a bit of a knife edge, but I think that was me playing it poorly rather than the spec having a problem. Raiding was rather hectic, but again I think that was me failing to really learn how to play. Discipline is a proactive spec, you have to be on top of things because you don’t have reactive healing. I do want to try it again some time, and with s4 coming up and having 2x priests at level 60, it might be the season for that!
That said, you can level open world as disc. It is slower than shadow, but my priest(s) have never specced shadow even once and that is a statement
Biggest let down; pugs don’t understand it and won’t work with it. They’ll happily die out of combat on grievous week, when what you need is for them to pull another pack so you can do some real healing. You kind of need a coordinated group to get the most out of a disc priest.
Holy Paladin
Honestly, never played! I find the race options a bit uninspiring. Although I may level one this weekend with the current xp buff. Already pushed a 2nd shaman up so’s to have one on each faction for the days when one side or the other is dominating random BGs.
Currently I am noticing a lot of complaints from holy paladins; they are worrying about mana, for the first time in several seasons. While there is some degree of pathos here, they also had over 2 years of ignoring mana entirely so maybe they’re just back to being a normal healer like the rest.
What I do know: completely melee focussed healer, even more than monk. You can’t get out of melee range and be effective, you cannot heal enough by standing back and casting.
Historically been strong in PvE and PvP, but that might have wavered recently. Suffers the same problem as disc priest; the group needs to coordinate with the paladin and not do things like squeeze them out of melee on Quaking week. They need to be there as much as the rogue and warrior do.
