Awake the Drakes

List of the different zones and instances where you're able to obtain the mounts:

[zone=4500] (Borean Tundra)
Both mounts drop (not necessarily simultaneously, but possible) from [object=193905] after you kill the one and only boss [npc=28859]. Drops from either difficulty.

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• [item=43952]
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• [item=43953]
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[zone=4493] (Dragonblight)
The mounts have a 100% chance to drop if you leave all three Twilight Drakes ([npc=30451], [npc=30452] and [npc=30449]) alive and then defeat the boss [npc=28860].

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• [item=43986] (drops from 10-man difficulty)
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• [item=43954] (drops from 25-man difficulty)
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[zone=4100] (Tanaris)
To loot this mount, you must clear the instance on Heroic difficulty and reach the NPC [npc=32273] within 25 minutes (timer available). Drops with a 100% certainty.

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• [item=43951]
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[zone=5892] (Tanaris)
This mount drops on all difficulties. Loot the [object=210079] after defeating [npc=56173].

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• [item=77067]
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[zone=2159] (Dustwallow Marsh)
Drops from [npc=10184] on all difficulties.

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• [item=49636]
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Dragonblight
Sold by [npc=32533] on top of the Wyrmrest Temple. Requires you to be exalted with [faction=1091].

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• [item=43955]
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Shadowmoon Valley
You need to complete the quest Lord Illidan Stormrage and be exalted with [faction=1015] in order to buy this mount from [npc=23464] (offers the quest Malfas the Purple Netherwing Drake). If you already completed the quest, the drakes can also be bought from [npc=23489] for 200 gold each.

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• [item=32860]
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Azure Drake: has a 4% drop rate from Alexstrasza's Gift in Eye of Eternity on 10m normal and strangely enough a 2% drop rate on 25m normal. Also a rare drop rate from an Unclaimed Black Market Container.
Bronze Drake: has a 100% drop rate off Heroic Infinite Corruptor a boss in The Culling of Stratholme so long as you can get there before the timer runs out.
Black Drake: has a 100% drop rate off Sartharion a boss in Obsidian Sanctum on 10m or 25m normal so long as you have all three semi boss drakes up when you kill her.
Blazing Drake: drops from the Elementium Fragment from Dragon Soul. It has a drop rate of 0.6% in 10m normal, 0.8% in 10m heroic and a 0.9% drop rate in 25m normal and 3% 25m heroic. There is also a very small chance to receive one from an Unclaimed Black Market Container.
Blue Drake: can drop from Cache of the Ley-Guardian in the Oculus but only at level 80 in a random heroic you can also receive it from Satchel of Exotic Mysteries 0.2% drop or the misty or savage satchels with a drop rate of 0.19 there is also a chance from the Unclaimed Black Market Containers with a drop rate of 0.7% or 0.9% depending on what one you get. O.o (Didn't know there was more than one to be honest.) But this mount is mainly contained within Alexstrasza's Gift from the Eye of Eternity with drop rates from 4% on 10m normal or weirdly 2% on 25m normal.
Onyxian Drake: has a 1.3% drop rate off Onyxia on 10m normal and a 1.6% drop rate on 25m normal. It may also be contained in Unclaimed Black Market Containers with a very low drop rate
Purple Netherwing Drake: You must reach exalted with the Netherwing rep in outland you can either receive this from the finishing quest or from buying it off Drake Dealer Hurlunk for 200g.
Red Drake: brought off Cielstrasza for 2000 (probably less once your exalted.) after becoming exalted with The Wyrmrest Accord.
Twilight Drake: has a 100% drop rate from Sartharion 25m normal with all the mini boss drakes up.

P.S If I have missed anything out or made any mistakes feel free to tell me and I'll get them changed when I see your comment. Most the drop rate info was found off Wowhead and is subject to change. Good luck on those mounts you lot, have fun!

In reply to Cerberus I'm uncertain now on the blue drake I thought it had been removed as well but from memory I had a friend who got it from a lvl 80 random heroic after I had thought it had been removed I will admit I could be mistaken though and as for the Blazing drake I have just checked the wowhead page for it and read that it does indeed drop on 10m heroic as there have been people in the comments who have received it. :)

(Updated drop rates for 2020.)

Well, I'm 99% done with this list but during the tail end of Cata content and during all of MoP, I have never seen Blazing drop at all...12-B...once and lost that roll to a former guildie and have not seen it again.

It also appears that the purple netherwing drake was removed due to the "wtf why is that there?" *edit: Sandstone is now being replaced with Purple Netherwing*

How to get:
Eye of Eternity raid:
Azure is a random drop from either 10 or 25m Malygos*
Blue is a random drop from either 10 or 25m Malygos**
* I have seen both drop at the same time on 10m
** For me, I had gotten Blue in Heroic Occulus when I believe lv 80's where able to get it
while in a random dungeon group.

Obsidian Sanctuary raid
Black is 10m Sartharion with all 3 twilight drakes alive*
Twilight is 25m Sartharion with all 3 twilight drakes alive*
*only one drake can be gotten from one run, you cannot do the Black run and then try
for a twilight run, this will require 2 resets

Wyrmrest Accord
Red is purchasable once exalted with Wyrmrest Accord in Dragonblight for roughly 1.6k gold (if your current guild you reside in has vendor discount)

Cavern of Time
Bronze is rewarded when you beat the timer and optional boss on Heroic Culling of Stratholm

Onyxia's Lair
Onyxian is a low drop from Onyxia's lair in Dustwallow marsh-can drop from 10 or 25m

Dragonsoul
Blazing is a very low chance when defeating Deathwing in 10m Dragonsoul

Shattrath City or Shadowmoon Valley
Purple Netherwing Drake is obtainable 2 ways
1) Once you reach exalted with the Netherwing faction located in Shadowmoon Valley in Outland and you can get him for free once you finish the quest line that takes you to Shattrath. Once there you can talk to Malfas-a purple netherwing and obtain him as a mount or
2) 200 gold from the drake vendor back in Shadowmoon valley

Most of these drakes, mainly raid and dungeon (not sure on Bronze) can appear in the Black Market Auction House in Pandaria for quite a bit of gold. It all depends on how populated the realm is.

Would like to apologize for my atrocious typing skills, I'm using an iPhone to text since my computer is down for now due to hacking-been without it for several weeks.