Some of you will remember this thread from Cata, so (having done a bit of soloing myself recently) I thought I'd get the MoP version going. I know there's some other threads scattered around, but I thought it'd be better to have all the kills and tactical discussions in one place. So I'll start with what I've done so far:
(updated 29/07/2013)
(current ilvl 544 with EM/PE, healing/feral spirit/LS glyphs, MoP agi flask, drums/runescroll III and 300 agi food)
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Vanilla/TBC: Everything.
WotLK 10: Everything.
WotLK 10H: XT-002, Iron Council, Hodir, Freya, Thorim (pre/post 5.0 + MoP), Mimiron, Vezax, Yogg+1, Algalon, Jaraxxus, Halion.
WotLK 25: Sarth+3, Malygos, all of Naxx, all of Ulduar except Thorim/Yogg (mechanics), all of WG, Onyxia, all of ToC, all of ICC except Gunship/Festergut/Blood Queen/Dreamwalker (mechanics).
WotLK 25H: XT-002, Freya, Vezax, Jaraxxus, Faction Champions, Twin Valkyr.
Cata 10: All of BWD, first 3 in BoT, first 4 in Firelands, first 4 in Dragon Soul.
Instances: Everything except Mannoroth in Well of Eternity (can't be in 2 places at once to move the RP along).
MoP Misc: All rares, all dinos apart from the big devilsaurs, Kork'ron commanders.
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Sept 2014 Update
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Apparently a lot more stuff can be soloed now, including a lot from Cata 25 and even some T14 10-man normals. Since I've not tried for so long, all I can suggest is have a go soon before 6.0 takes our soloing ability away. Some quick tips:
Glyphs: Healing Storm, Lightning Shield, Feral Spirits
Talents: SBT/Earthgrab/Persistence/EM/AG/PE
Stats: Agi >>> Haste > Crit >>> Mastery
Consumables: Drums of Forgotten Kings, Runescroll of Fortitude III, agi flask & food
Keeping yourself topped is more important than doing dps, unless there's a specific nuke phase or dps race.
Spend all MW5's on heals unless you're topped.
Treat EM, wolves and hero as defensive CD's, due to increased MW procs. (unless there's a dps race)
Use AG and ascendance together if possible. (especially if you have HC assurance trinket)
Don't forget primal earth ele's channel, it's the best defensive CD we have due to it's length. (but also note that it can't be used when kiting)