Music.
Discogs API is a free Music API that requires OAuth authentication, is served over HTTPS and has unconfirmed CORS support.

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Yes. The Discogs API is listed in the Music 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.
Yes. The Discogs API uses OAuth authentication, so you need to register for credentials before making requests.
CORS support for the Discogs API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Discogs API is served over HTTPS, so it's safe to use in production.