you can solo anytime you want in this game. unfortunately, in order to lvl up its practically required that you pt. there are maybe 4-5 classes that are able to solo for xp. the other 15-16 jobs are forced to join a pt for maximum xp. waiting for a party can be the most frustrating thing in FF.
now in terms of missions/NMs certain classes solo better than others. most mage jobs are able to solo quite well. rdm, blm, blu all do very well. any job with pets can solo well, bst, smn, pup, even drg/mage. most jobs w/o a mana pool will have the hardest time soloing in this game.