I'm not sure what kind of content you'd want to convey, but I'd start with RFC 5785 Defining Well-Known Uniform Resource Identifiers[1] (which defines /.well-known/), and perhaps move on to RFC 6415 Web Host Metadata[2] (which defines host-meta.json, providing a set of links).
RFC 6415 isn't all that popular, but it's job is simple enough- provide a list of links to resource types. Each kind of service ("search") probably *should* in any decent web-y spec have a IANA Media Type/MIME Type, and RFC 6415 is a direct way to enumerate all services by their MIME type. Specifically look to the JRD section for the JSON form. This is suitable if all you want is a media-type (here, the media being perhaps a service of a certain type) and a URL to start talking to that media type.
OpenAPI is around but I'm not aware of any facility for example for Alice and Bob to both advertise that they have implementations up of Chad's Search Service interface. It seems more catered to describing the shape of your own API, but free from external context. I could be off base here, and if so, OpenAPI would make a great choice simply because it has wide-spread adoption.
Last, there's a bunch of different ways one could discover an EntryPoint[3]. Since it's JSON-LD, one can embed EntryPoint data-islands or microdata onto any HTML page, identifying potentialActions that are available at that point. There's already a strong set of common well known actions that pages can offer, from ViewAction to ShareAction & anyone wanting to create new action types has only to define a namespace for their actions.
[1]
https://tools.ietf.org/html/rfc5785[2]
https://tools.ietf.org/html/rfc6415 [4]
https://schema.org/EntryPoint