Any class can successfully solo PvE content at a character’s level or below, as long as the content doesn’t require a group at close to the character’s level, or specialized skills like tanking.  Some classes are easier, some harder, and some content will only be possible with certain classes or specializations.

Some sites and forum posts suggest only the pet classes for soloing, because you have an effective tanking partner:

• Hunter
• Warlock
• Death Knight

Usually the Druid is added to the list because of its transformative abilities to switch from Rogue-like DPS (Cat form) to Tank (Bear Form) and to self-heal without changing form.  Use a Night Elf and you can disappear and drop combat then switch quickly to Flight form and fly away. Prowling invisible is a fast way to zip through old dungeons for mount runs etc.

Solo dungeon running of older content is often easier on one of the tanking classes: DK, Protection Paladin, Warrior, Guardian Druid, Brewmaster Monk.  Best choices in many old dungeons and raids are those with lots of good AoEs, such as the Protection Paladin.

Some old dungeons have mechanics that best fit a specific class and spec.  E.g. Cho’gall last boss of Cata’s Bastion of Twilight has a very fast timer which gives you only a couple of shots to kill him before he resets.   A BM Hunter with a high DPS pet like Portent (Quillen) plus traps and Barrage can do it easily with the right rotation.  E.g. the Ichoron boss in Violet Hold needs quick movement to the Hold’s defense buttons and having strong AoE also helps, so a high-level Mage works well.

I’ve soloed various content with these classes and specs: BM Hunter; Survival Hunter; Druid (Guardian spec in Bear and Cat forms); DK (interestingly not my fave, my DK is not as strong as other tanks IMO); Prot Pally; Disc Priest; Enhancement Shaman; Frost Mage.