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I'm building a new character for our weekly game and a build-around that I think could be fun is to optimize for the number of times I roll on the Wild Magic Surge table while dealing respectable damage. I'd like to do this via a multiclass: Sorcerer (wild magic origin) 1 / Wizard (school of evocation) 10. My DM has already informed me that any time the they may have me roll on the Wild Magic Surge table, they will choose to do so.

The wild magic sorcerous origin states that "the DM can have you roll a d20 immediately after you cast a sorcerer spell of 1st level or higher" (emphasis mine). The Empowered Evocation ability of the school of evocation states that "you can add your Intelligence modifier to one damage roll of any wizard evocation spell you cast" (emphasis mine).

Would it work within the rules (without asking the DM to create house rules) for me to learn Magic Missile through my Sorcerer Spellcasting feature, and to also choose Magic Missile as a spell to add to my Wizard's spellbook and prepare it using my Wizard Spellcasting feature, such that when I cast Magic Missile it gets both of those abilities applied?

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A spell is a Wizard spell or a Sorcerer spell, not both

This does not work, RAW. While your spell slots are shared between classes, any given spell is cast via one class or the other, not both. The most obvious way in which this matters is your spellcasting ability; when casting as a Wizard, you use Int, while when casting as a Sorcerer, you use Cha. A single spell can't be cast with both. This is supported by the multiclassing rules (emphasis added):

You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class. ... Each spell you know and prepare is associated with one of your classes and you use the spellcasting ability of that class when you cast the spell.

In short, you can't have it count as both. This gives you the option to cast as a Sorcerer (using Cha, and maybe getting a wild magic surge), or as a Wizard (using Int, and adding it to damage, and definitely not getting a wild magic surge). But it's an either/or choice, not an and/or choice.

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