The W3C Data Exchange Working Group has issued first drafts of two potential standards relating to profiles.
The first is a small ontology for profile resources. This ontology is based on the case where an application profile is made up of one or more documents or resources, such as both human-readable instructions and a SHACL validation document. The ontology links those into a single graph called a "profile" and states the role of each resource.
The second is a companion to an (in-progress) IETF proposal that would support requesting web resources that are conformant with a profile.
If you have an interest in profiles please look at and comment on these proposals. Instructions for commenting are included in the documents themselves.
(on behalf of Karen Coyle, Co-Chair of the W3C Data Exchange Working Group)