What are the trade-offs between different methods of constructing API URLs: subdomain vs. subdirectory and versioning?

We have a web application with a domain name of example.com. Now we want to extend a part of this application as a REST API, and we are debating on the best URL pattern.

We could use the URL pattern api.example.com or example.com/api. What trade-offs are there to consider, if any?

Additionally, what trade-offs are there regarding methods of API versioning? It could be done via the URL (v1.api.example.com, example.com/api/v1, or some strange mix v1.example.com/api or api.example.com/v1). Alternatively, it could be done through the use of HTTP request headers (or otherwise)?