CC0 is a public domain dedication, with fallback license for jurisdictions where public domain dedication is not possible. The public domain dedication part is not a license (this was a large part of the legal criticism of the awfully-named Unlicense: that it tried to be both a public domain dedication and a license, despite the two being incompatible), and so cannot do anything with trademark or patent rights.
If you want to cover trademark or patent rights, you’ll have to use a public-domain-equivalent license instead of a public domain dedication. This is where things like 0BSD come in. (With this latest thing, I think I’m going to change my primary recommendation in https://chrismorgan.info/blog/unlicense/ to 0BSD rather than CC0.)
CC0 is a public domain dedication, with fallback license for jurisdictions where public domain dedication is not possible. The public domain dedication part is not a license (this was a large part of the legal criticism of the awfully-named Unlicense: that it tried to be both a public domain dedication and a license, despite the two being incompatible), and so cannot do anything with trademark or patent rights.
If you want to cover trademark or patent rights, you’ll have to use a public-domain-equivalent license instead of a public domain dedication. This is where things like 0BSD come in. (With this latest thing, I think I’m going to change my primary recommendation in https://chrismorgan.info/blog/unlicense/ to 0BSD rather than CC0.)