You should enter it under your spouse.
When you first enter business information under Self-employment income and expenses, TurboTax will ask you Who did this work or owned this business?
If your spouse did the work, you should indicate that it is his business. This is because Self Employment tax has to be credited to an individual.
If you and your spouse own a business together, you’ll need to split your business income and expenses between the two of you according to ownership percentages, as if they were two separate businesses.
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Could you please clarify just a bit. You say the business is in your name. I assume that means you are the only owner? And, if your company generated 1099's for any and all of those who provided services to your company, you should only need to generate one 1099 to his name (unless he did these services for your company using several different companies that he may own).
Further, assuming that I have the basic facts straight per above, the 1099's that your company generated typically don't need to be "signed" by anyone. But they do need to clearly show your company's name on each one generated, of course along with the name, social, address, amount, etc. for the non-employee (1099) workers.