I expect you know what a swan is and can probably see where the term swan necked comes from. The birds’ graceful progress across the water would account for the phrase swanning about too (I swan about a fair bit, though not usually in a graceful manner. )
A swan song is a person’s last work or act before death or retirement. I’m not sure why a large white water bird of the genus Cygnus should be associated with that, or with a form of diving, but it is.
Did you know that swan can also refer to a poet? And can you guess who has been referred to as the swan of Avon?