PHB p. 72 (and others)
Dueling
When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.
PHB p.147
Thrown. If a weapon has the thrown property, you can throw the weapon to make a ranged attack. If the weapon is a melee weapon …
So a Handaxe is a simple melee weapon with the thrown property so, as written, it can benefit from Dueling whether used for a melee or ranged attack because it remains a melee weapon irrespective of how it is used.
Conversely, it cannot benefit from the Archery Fighting Style because it is never a ranged weapon even when making a ranged attack.
Where it gets really interesting is in the interaction with Two-Weapon Fighting (p.195)
When you take the Attack action and attack with a light
melee weapon that you’re holding in one hand, you can
use a bonus action to attack with a different light melee
weapon that you’re holding in the other hand. You don’t
add your ability modifier to the damage of the bonus
attack, unless that modifier is negative.If either weapon has the thrown property, you
can throw the weapon, instead o f making a melee
attack with it.
So, with 2 Handaxes and I throw the first, because it is a light weapon, I can use Two-Weapon Fighting to use the other (either for melee or ranged) and benefit from Dueling because I am now "wielding a melee weapon in one hand and no other weapons."
Am I reading this right?
Best Answer
I'll get the simple bit done first - you're right, nothing in the Thrown property turns a melee weapon into a ranged weapon. It's a melee weapon you can use to make ranged attacks, so the Archery Fighting Style, which says that
can never apply to it. It can apply to darts, because they are thrown ranged weapons. This was also confirmed in a recent Sage Advice column:
Now for the Dueling Fighting Style. Here, I have to disagree with you - when you roll damage for a thrown weapon, you aren't wielding that weapon in one hand. You were wielding it, then you threw it, then it hit. If you were still wielding it when it hit the target, it would be a melee attack. (Unless you threw it, then teleported across the battlefield and caught it right when it hit them, which sounds cool, but doesn't seem particularly useful.)
"Wielding" is a fairly nebulous term, so there is room for interpretation here, but personally I'm inclined to think that "wielding [...] in one hand" pretty clearly requires you to be holding something in your hand.
On the other hand, Crawford says that the Dueling Fighting Style does work with thrown melee weapons, and he's the authority on these matters.