Edit (Senen): I try to maintain a spreadsheet up to date. If a kill is missing from it, please provide some information about it.
Molten Core (lal)
BWL (if I get help with Razorgore, then I get the rest. Changes in 5.1, soloed it on PTR)
AQ40 - All except Vicidous (or whatever it's called) and C'thun because of mechanics. Vicidous may change in 5.1, but I doubt it. You just need to go with a frost dmg class to kill it for the pet.
"The Eye" - Kael'Thas (for mount) not tried the others as they got nothing that motivates me to kill them
Magtheridon (duh! to easy to even mention)
Karazhan (chess is easy...if you got luck with you!)
Hyjal Summit -Rage Winterchill, not tried any other
SSC - Never tried
Black Temple - Never tried solo
Sunwell Plateu - Trash killed me already, so I guess I wont be able to kill anything in there
Naxxramas 10 - Full clear as of tonight (Instructor Razuvious was actually the hardest of them, if you want my tactics just ask)
OS10 3 drakes
Malygos 10 (doubt there's time to kill 25 solo)
Onyxia 10 and 25
TotC - Not tried, will do tonight hopefully
Ulduar 10 - FL hardmode (all towers up) and all bosses until Mimiron (except Thorim). With some tweaking then it should be possible to do hardmodes on some of them. I got it for Hodir at least.
ICC 10 - Lord Marrowgar and Lady Frost, Gunship stopped me to go further. Will get help for it so I can try later bosses.
Pretty much all HC dungeons
Nothing so far (except rares)
Probably forgot some.
Do you know some challenging ones that don't require tank-gear? Is that last drake soloable that was added in LK?