What is a word for something that someone is particularly known for. e.g. a song that a singer always is asked to sing
Signature is the answer. Both as a count noun and an adjective.
[count noun] something (such as a quality or feature) that is closely associated with someone or something — usually singular
Her latest movie includes the humor and fast pace that have become her signature.
[adjective] closely associated with someone or something : making a person or thing easy to recognize
the chef's signature dish
“Respect” is her signature song/tune.
The director's new movie is a return to his signature style.
Trademark: A distinctive characteristic by which a person or thing comes to be known.
E.g. (Lifted from @ermanen)...
...1-Her latest movie includes the humor and fast pace that have become her trademark.
2-The shuffle and snicker that became the comedian's trademark.
*Though "signature" would be the first thought for something that someone is particularly known for.*
For your particular example, I think the term "magnum opus" is appropriate. According to Merriam-Webster the definition of magnum opus is as follows.
magnum opus - noun - a great work; especially : the greatest achievement of an artist or writer.
the typical or highest example of a stated quality, as shown by a particular person or thing: Even now in her sixties, she is the epitome of French elegance.
src: Cambridge online
Also the characteristic would be antonomasia
: the use of a proper name to designate a member of a class (as a Solomon for a wise ruler); also : the use of an epithet or title in place of a proper name (as the Bard for Shakespeare)