*Trial of the Crusader
Northrend Beasts: World first hunter solo: Caribald vs Northrend beasts (heroic) - YouTube
Lord Jaraxxus: World first hunter solo: Caribald vs Lord Jaraxxus (heroic) - YouTube
Twin Val'kyr: Durendil vs Twin val'kyr - Heroic - YouTube
Anub'arak: Durendil vs Anub'arak heroic - YouTube

*Icecrown Citadel
Lord Marrowgar: YouTube - ‪World's First Lord Marrowgar ICC10 HEROIC Solo‬‏
Lady Deathwhisper: Durendil vs Deathwhisper heroic - YouTube

Festergut: World first solo (hunter): CaribaLd vs Festergut (10 man heroic) - YouTube
Rotface: YouTube - ‪World's First Rotface ICC10 HEROIC Solo‬‏

Sindragosa: World first hunter solo: CaribaLd vs Sindragosa (10 man heroic) - YouTube

Expanding on my (much) earlier post on 5-man level 80 heroics:
Round 2: Fairly typical MM-DPS talent (no spirit bond - I picked that up later to beat Saphiron), Buffed with the kings-drums, alchemy permanent flask and random useful food. Averge Illvl just about 346.

Forge of Souls ends up not being so bad even on heroic - the trash is significantly harder than the bosses, but doable - or avoidable either with Camouflage or run sufficiently far past + feign. I can't see how to get Bronjahm's achievement. The devourer of souls achievement looks doable (his phantom blast appears to have a shortish range and not target the pet? Or at least he only attacked me with it a handful of times when I got *way* too close).
Utgarde Tower was unremarkable (forgot about it last time); Svala is the most annoying (pet+trap don't seem to get through the ghosts very fast). Not sure if her achievement will be doable - I was happy just to get her our of the way. The other achievements are easy.
Moragg in Violet Hold is actually not so hard, if you do what every melee character would probably do automatically: Stay behind him! His really nasty attack seems to only work in front of him. Evenso, his damage is respectable, but nothing that will overwhelm you. Xevozz is painful but doable - there's not much to say about him; probably much easier if you're good at micromanaging your pet's movement, which I'm not. Still managed it after a few attempts (dead pet though).
As I expected earlier, the Tribunal of Ages is very much doable as MM due to a nice amount of self-healing. The achievement is tricky; the number of adds running all over everywhere gets large in the last 15-30 seconds; it's easy to not get the attention of one. Be a little conservative with focus to always have a little to distract naughty adds from Brann.
Both Zombiefest and the timed run in strat are doable; for Zombiefest, I put my pet away (I don't think they have any non-damaging AOE aggro?), and made liberal use of Posthaste-buffed disengage when collecting zombies; 50+ zombies do end up hurting. Engineer is nice for the cardboard assassin The timed run boss is really nasty - I failed 3 times because of underestimating him; he again has a stackable %-life-lost-per-3-seconds debuff. Take a breath before engaging, get out a DPS pet and pre-buff with the pet +5% crit or whatever you chose, and nuke for all you're worth. MM with self healing and double rapid fire may be necessary (at least with my gear level - 346ish). On the succesful attempt I took almost exactly 20 seconds - so a double RF - and was still down to something like 25% life.
Oculus is easy until Eregos, but beating him seems a lost cause; red is the only viable dragon (bronze does next to nothing with no other DDs there, and green doesn't have enough GCDs to dispense enough DoTs to keep pace with spawning mini dragons), but he still handily outdamages me. On foot I couldn't get my pet to engage at all; maybe there's a cool positioning trick, but I couldn't find it.
ToC5 champions seems equally un-doable; on foot the trash is easy, but the champions themselves constantly push you and your pet over (pet tanking doesn't work since they ignore it when they've pushed it over). I can't imagine riding the horses solo will work, but I guess I should try.
Hadronox Denied is just a question of triggering her to despawn and setting up to nuke hard when she respawns. Just a race to kill her before her adds start healing her faster.
Anub'Rekhan is still fairly nasty; I did get his achievement this time around, but with all the adds it was still touch and go at times, even with MM healing.

EDIT after Haero91 pointed out what Master's Call actually does (I had previously completely misunderstood the ability description):
Dual Master's Call with and Rapid Fire (via Readiness) was enough for Intense Cold, even with my equipment. As Haero91 notes, PvP trinket can get you another escape, and/or I believe the living action potion will get you out of yet another.
On the Rocks: Nuke as hard as possible until the first ice cube (stand so that the skeletons come to you); when the ice cube ticks off naturally (Master's Call doesn't get you out - "Can't do that while stunned") mend pet, cobra shot, feign (skeletons leave/despawn), and let your pet knaw on him for a while. My pet (turtle) easily got him into the kill shot zone before its health began to look even remotely threatening. I got up with master's call (not sure if the 4 second immunity actually helps here, but it was worth a try) and nuked him the rest of the way. Got another ice cube just as he died.

I think I'm running out of soloable achievements. The open ones:
Less-Rabi may be doable with some serious nuking, silencing shot + pet interrupt + readiness-silencing shot, but I've so far failed to get the timing right - it'll be very close certainly. I'm an engineer and so would have the bolt gun interrupt, but I think pulling off the three more readily available interrupts right should be enough.
Share the Love may be doable if swapping pets (or just re-summing one?) counts - but that means we need to tank him ourselves a fair bit, and he hurts even at 85, particularly if we're going to be seriously extending the length of the fight. I'm skeptical.
Watch him Die *MIGHT* be doable with really hard nuking, but the damage on your pet is high - I think I'll need some serious equipment upgrading before I stand a reasonable chance.

Random aside: The Lurker Below can be fished with a skill of 1. I got him on my first cast with the standard line, no lure, nor any other buff.

Except the fact that this video above shows a normal mode solo. In Heroic mode with "no gear scaling"-horses, your mount will be obliterated in seconds in the champions fight, and the spawn of new horses takes the roughly 2x more time to spawn again.

Video: YouTube - ‪Durendil vs Hodir‬‏

So, I solo'd Hodir. I used magical dps strategy, since physical dps strategy requires much better gear (the guy used like every buff and potion possible, and stayed in aspect of the hawk for all the fight).

My strategy, even if it will probably become obsolete in a few gear levels, was to dps magically, by spamming cobra and arcane shots.

THE FD ISSUE: if you FD, Hodir despawns immediately. However, if he is casting Flash Freeze, he will despawn later. So if you want to reset your aggro and go all-out dps, FD while he is casting it, then attack him before he finishes his cast.

THE NPCS
All NPCs have like 1K5 dps.

Mage: - frees other NPCs from flash freeze
- puts down fires: for the magical dps strategy, they are the heart of the fight.

Druid: -puts light columns: big haste boost while inside.

Priest: keeps NPCs alive, unless you spend too much time making them eat icicles and freezes. Sometimes frees you from a freeze.

Shaman: puts a cloud that doesn't buff you but buffes nearby players (bad shamy)

SINGED: when you hit with cobra shot, arcane shot or chimera shots (since it reapplies serpent sting), and you stand near a fire, you have a chance to apply singed, dealing extra fire damage (3K to 4K5, depending on stacks). Singed increases magical damage dealt by 2% per stack, up to 25 stacks! To down Hodir, he must destack as least as possible.

HODIR:
IMPORTANT: Flash freeze not the same thing as Freeze (root).

Icicles: avoid those. They always lock on your current position. They bump, so it's possible to move NPCs around with them.

Biting cold: debuff that gains in stacks when you don't move, and loses stacks when you move. Does not affect people near fires.

Freeze: roots you in place. Use Master's call or eat a lot of icicles. Does not affect people near fires.

Frozen blows: Hits your pet hard. Use CD and heal him.

Flash Freeze: the heart of the fight. He will make a big icicle fall. Avoid it. If you are bumped, don't bother running: jump-disengage or be frozen. You must get on the snow to avoid being frozen.
The bad thing is that it flash freezes the mage, and extinguishes the fires: his stacks might disappear... To avoid him losing his stacks, there are two methods.
-If you are lucky, and you stand near the mage, but not too near, but not too far, then he won't get bumped but will stand on the snowdrift (very rare). Then he will put down a fire, and Hodir won't destack. However, Hodir will freeze you: use Master's call, then go near the fire, and spamclick you magical abilities.

-The usual way is for the mage to get Flash freezed.
Go all-out on magical attacks before he casts flash freeze, as to reset the debuff. Then build up focus
Once the mage is freezed, go all-out dps to free him. Hodir will freeze you : use master's call. Use cobra shot to build up focus. Watch the mage closely: as soon as he puts down a fire, run near it, and go all-out with magical attacks before the debuff expires. If all goes well, you should reset the debuff. There is a bit of randomness associated with this strategy.

Continue to dps until he hits 100K: use rapid fire at the start, as to build stacks quickly and use 6 rapid fires in the fight.