The spec:
https://www.ietf.org/rfc/rfc4918.txt
A very long spec, especially for one that seems relatively basic. There also seems to be a lack of entry level information so I've mostly pieced together things and just hope things work.
Upload a file, -T is the PUT method:
curl -T file.txt http://user:pass@localhost/webdav/file.txt
Get a file:
curl -X GET http://user:pass@localhost/webdav/file.txt
Delete a file:
curl -X DELETE http://user:pass@localhost/webdav/file.txt
Get a listing of the directory:
curl -X PROPFIND http://user:pass@localhost/webdav/file.txt