Evil Insults.
Evil Insult Generator API is a free Open Source Projects API that requires no API key, is served over HTTPS and supports CORS, so it can be called from the browser.
Details from evilinsult.com.

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Word-finding query engine
The Evil Insult Generator API is provided by Evil Insult Generator (evilinsult.com).
Yes. The Evil Insult Generator API is listed in the Open Source Projects category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
No. The Evil Insult Generator API requires no authentication, so you can start making requests without signing up.
Yes. The Evil Insult Generator API supports CORS, so front-end JavaScript can call it directly.
Yes. The Evil Insult Generator API is served over HTTPS, so it's safe to use in production.