Divine Smite: Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one paladin spell slot to deal radiant damage to the target, in addition to the weapon's damage.
I'm multiclassing into Paladin/Warlock/Sorc, and I'm wondering: can Divine Smite only be used in spell slots gained specifically by taking a level in Paladin, or can all available spell slots be used?
The errata to the Player's Handbook cleared this up:
Divine Smite (p. 85). You can expend any spell slot, not just a paladin spell slot.
You can use any spell slot to Divine Smite, regardless of where it comes from.
As per the multiclassing rules, all of the other spellcasting classes get their spell slots lumped together for the purposes of casting spells (see the Multiclass Spellcaster table on PHB page 165). A Sorcerer or Wizard spell slot is functionally no different from a Paladin spell slot of the same level. We can therefore conclude that Warlocks are the only outliers, due to their unique Pact Magic spellcasting. The relevant question becomes whether or not you can use your Warlock slots for your Paladin smites.
Yes, you can use your Warlock spell slots to fuel the Paladin's Divine Smite ability.
Jeremy Crawford, the singular official source for all D&D 5e rules clarifications, has stated unequivocally that Warlock spell slots are usable with the Paladin's Divine Smite.
Also, as Miniman helpfully pointed out in his own answer, the Player's Handbook errata also states that you can use the spell slots gained from levels in other casting classes with the Paladin's Divine Smite.
When you multiclass your spell slots change to match the table (Pg 165 PHB). Those spell slots are shared between all your spell casting classes and can be used in any combination. You would then have all those slots available for Divine Smite.
Example.
2 Paladin/3 Sorc/2 Warlock
Gives you 3 spell level for Sorc, and 1 for Half Paladin level. This makes you spell level 4. Giving you 4 first level slots and 3 second level slots. Your paladin levels contribute to your spell slots so therefore you could use any of those for Divine Smite.
You would be able to use the warlock pact slots to cast Sorc spells but as Paladin Levels do not contribute to your pact slots only spell slots they would not be usable for Divine Smite.
Most other spell casting refers to the use of Spell Slots in general and not a class specific spell slot. This case of Paladin Spell Slot leads to the answer above.
Depending on DM interpretation some DM's might rule that Warlock pact slots can be used and others would rule otherwise. If the DM lets you use warlock pact slots then the Divine Smite would be used at the level of the spell slot.
The paragraphs on PBH 164 and the Table on PHB 165 explain it. All spellcaster classes contribute. 1:1 for Primary Spell Casters (Wiz, Sorc, Cleric, Druid, Bard), 1:2 for Secondary Spell Casters (Paladin and Ranger) and 1:3 for tertiary spellcasters (eldritch knight and arcane trickster). Plus Warlock Spell Slots.
All slots can be used for any class's spell.
There is no real spell slot advantage to multiclassing. It doesn't let you cast more spells, it just lets you choose from a larger pool of spells. Except in the case of the warlock where you actually do gain 1 or 2 additional spell slots or even 3 or 4 if you are higher level warlock, but then again, you are trading that for levels/slots in other spellcasting classes, so it may not be a real advantage. I'll have to spreadsheet this to see.
