Yes, you can. I'm not sure about other classes though, but Hunters can do it relatively easily. It takes some kiting here and there on big pulls, but generally you can have an endurance or ferocity pet do the tanking for you and then you just blast shit off. I solo'd shadow labs just fine on my Survival last week for the socketed boots and bracers from the quests. The only annoying thing was Grandmaster Vorpil, but because the fight is kinda ass rather than hard (the whole voidwalker explosion thing + "haha I'll teleport you" can be somewhat triggering). Murmur was just a tank 'n' spank with careful usage of mend pet to keep the pet alive and tanking. If you're having trouble maintaining your health pool up, you can also stick to a Ferocity pet for the 10% leech. It may not seem like a lot, but Survival specifically benefits a lot from it. You'll heal off of your dots, Raptor Strike spam AND your mastery on top of that. It is particularly useful when dealing with large packs. It also helps to manage the health of the pet as Kill Command will heal it.
I'd imagine Fury Warrior makes for a great solo class once geared up because they scale very well with gear and double crusader procs offer good survivability + nice burst windows. Add to that the fact they can cleave every skill with whirlwind now, and trash packs can likely be cleared with relative ease.
Generally though I would say every class has at least one spec that can solo dungeons, albeit ones slower than others of course. It will all come down to the gear you have and your patience to deal with the potential need to kite mobs and stuff.