Can the benefits from dueling/great weapon fighting styles apply when using a lance while mounted/unmounted respectively?
Are Lances a DEX or STR weapon in 5e?
What other perks are provided by wielding a lance?
Melee Weapon (Martial)
Damage Type: Piercing
Item Rarity: Standard
Special: You have disadvantage when you use a lance to [a]ttack a target within 5 feet of you. Also, a lance requires two hands to wield when you aren't mounted. [See PHB pp. 149-150]
As we can see above, the lance does not have the two handed or versatile properties; one of which is required to benefit from the Great Weapon Fighting fighting style:
When you roll a 1 or 2 on a damage die for an attack you make with a melee weapon that you are wielding with two hands, you can reroll the die and must use the new roll, even if the new roll is a 1 or a 2. The weapon must have the two-handed or versatile property for you to gain this benefit. - PHB p.72
You could use Two-Weapon fighting instead if you are mounted and take the Dual Wielder feat, though.
But only while you're mounted; this weapon's special property states that when you aren't, it "requires two hands to wield". Because of this, Dueling, which requires that you use the weapon in one hand, can only apply while you are mounted:
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 72
All melee weapons in 5e are STR based unless they have the finesse property (in which case you can use either STR or DEX). While there may be ways of getting around this, a previous edit suggested you were planning a fighter or paladin, and the linked solutions wouldn't work there.
If you're thinking of playing a mounted melee character, the lance gives you reach without the two handed requirement (if you're mounted) and does more damage than any other weapon with that property (d12 is the highest damage dice available for official weapons). This means if you're playing, say, a mounted fighter, he can wield a d12 weapon in one hand with a shield or secondary weapon in the other (if you take the dual wielder feat - Lance doesn't have the light property), which is pretty impressive! So in Summary:
Reach - esp. on a mount, you can run around without taking Attacks of Opportunity, and you can get at monsters blocked by another character or monster.
d12 damage - the highest available from an official weapon.
A free hand (if you're mounted) meaning a shield or (if you take Dual Wielder) a second weapon. Imagine 2 lances!
No other weapon gives you all three of these at once.
However: Within 5 feet attacks are at a disadvantage, so unless you have some way of keeping your foes outside of normal melee range (5 ft), if you want the lance as your primary weapon you're going to be either missing a lot more or provoking attacks of opportunity almost every turn, which sucks a fair bit. This is the price you pay for the above advantages, though Dueling could help offset it if you don't intend to put anything in your other hand.
Opportunity attacks could be avoided by using the Mount's limited action, as the mount is the one doing the moving. (Thanks to @guildsbounty for pointing that out!)
In 5e the lance is a martial melee weapon with two properties: reach and special. As it lacks the finesse and light properties, characters use their Strength bonus* to calculate their bonus to attack with it, and cannot use it for two-weapon fighting without taking the Dual Wielder feat (PHB chapter 6).
While it lacks the two-handed property, the specific special rules for the lance state that if used while not mounted it requires two hands to use, so even with the feat it could only be used for two-weapon fighting from horseback.
The reach property means you may attack foes within 10 feet instead of the usual 5, but its other specific special rule is that it imposes disadvantage if attacking a target within 5 feet, so it’s really not designed for regular D&D melee combat.
Best case for two-weapon fighting with the lance might be a mounted character not using all their movement, attacking a foe 10 feet away with the lance, then moving the horse within 5 feet to attack using a bonus action with a one-handed melee weapon in their other hand. (The two-weapon fighting rules do not say you cannot move in between the Attack action and the bonus action to attack with the second weapon.) This would require the Dual Wielder feat, as above.
*A few specific character types can get around this, though the only one I know is a Warlock with the Hexblade Patron from Xanathar’s Guide to Everything. They could use their Hex Warrior feature to allow them to use Charisma when using a lance, whether a regular one or one summoned with the Pact of the Blade feature.
If you use a Lance with a Kobold, you can use your lance without disadvantage at close range, since your mount counts as an ally.
Additionally, the Kobold's small size and light weight means that you can use mounts of smaller size. My Kobold Cavalier, for example, wields a lance and rides a Giant Weasel.