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I understand there are specifications for public-key fingerprints that use modern hash algorithms like SHA2. What are they?

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Any secure hash algorithm can be used to generate a public key fingerprint - your choice depends entirely on which hash is expected by the other party.

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I recall reading some standard about taking the SHA2 hash of a DER representation of the private key, and a standard name for this kind of fingerprint. – joeforker Feb 8 at 13:57
Yes I did see this when looking around for an answer for you - I'm not overly familiar with fingerprinting but is this the document you're looking for? tools.ietf.org/html/draft-ietf-smime-sha2-11#ref-SHS – Alex Jeffrey Feb 8 at 14:04

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