Schedule C vs. Schedule E -- my wife received a 1099 from a subtenant for subleasing office space she doesn't own. Which schedule should I use, C or E, for this income?

You should report the income directly on Schedule E because this is passive income. ‌You will be able to deduct the costs related to subleasing the space directly on the expense section of the Schedule E.  You wouldn't be able to deduct depreciation since your wife doesn't own the office space.  See the below TurboTax help article for steps on how to enter the rental income.  

Where do I enter income and expenses from a rental property?

Regarding the form that was issued to your wife, the rental income has been reported incorrectly.  The payer should issue a corrected Form 1099-NEC changing the amount to $0.   They need to issue a Form 1099-MISC reporting the rent paid.