In theory, fiancés have the right to work in the U.S. during the 90 days they are allowed to be in the U.S. to marry a U.S. citizen. In practice, taking advantage of this right is more difficult than it sounds—close to impossible, in fact.
As a K-1 visa holder, you can work in the U.S. only if, after entering, you apply for and receive a work permit. This is known by U.S. Citizenship and Immigration Services (USCIS) as an Employment Authorization Document (EAD). It is a small plastic card with your photo on it.
For application procedures, see below; but first, keep reading for the reasons you might not want to apply.
The difficulty with applying for a work permit is that the USCIS Service Centers routinely take from three to five months to issue them. This means that your chances of receiving your work permit while you are still eligible for it, as the holder of a fiancé visa, are slim to nonexistent.
You would likely be better off marrying as soon as possible and focusing on preparing your adjustment of status (green card) application.
As part of your application packet, you can include a Form I-765 requesting a work permit. In fact, under the fee terms applicable in early 2020, you won't have to pay anything extra for it if it's included with your adjustment packet—which is better than applying while you are still only in fiance visa status, in which situation you would have to pay a separate USCIS fee. (But note that USCIS is planning to change it's fee structure soon, so this might change.)
Once you submit your adjustment of status application, you will still probably wait anywhere from 60 to 90 days for your work permit. You can check the USCIS Processing Time Information page of the agency's website to find out how long other I-765 applications are taking at the time that you file yours.
You'll need to know which Service Center is processing your application. Look for this information on the receipt notice you received from USCIS when you filed your I-765. That is where your work permit application will be processed, despite the fact that you will be interviewed for your adjustment of status application at a local USCIS office.
On the USCIS webite, look for the processing time for an I-765 "Based on a pending I-485 adjustment application [(c)(9)]." Again, if it's three months or more, there's no point in applying.
If you've found that the processing times have speeded up and so you do want to apply for a work permit, use Form I-765, Application for Employment Authorization, available for free download from the USCIS website.
For general instructions on how to prepare the form, read Filling Out Form I-765, Application for Employment Authorization.
For Questions 24 and 25 of Part 2 (5/31/2018 version of the form), your answer will most likely be K-1 fiancé.
Part 2. Question 27: Your eligibility category is (a)(6). Leave the third set of parentheses blank.
When finished, mail the form, the fee, two passport-style photos, and proof of your fiancé visa status (such as a copy of your approval notice, the stamp in your passport, and your Form I-94) to the appropriate USCIS Service Center, as indicated on the USCIS Web page for I-765 filing addresses.
You may pay by money order, personal check, cashier’s check, or by filling out and submitting Form G-1450, Authorization for Credit Card Transactions.
After sending your application in, you'll wait. And no matter how long USCIS takes to approve it, the EAD will expire at the same time as your K-1 status.