The Danish Hollywood actor Jean Hersholt (1886-1956) was an avid collector of Andersen editions, and among other things he translated Hans Christian Andersen's fairy tales and stories in the excellent edition The Complete Andersen (six volumes, New York 1949) - which you may read on the web site of the Hans Cristian Andersen Center. There is also a search engine for the tales.

As the title conveys, The Complete Andersen is a rather extensive collection of Andersen's tales - and this web version is the most comprehensive edition of Andersen's fairy tales in English (American) on the internet, where Mrs. Henry B. Paull's old (1867-72), victorian translations has been dominating.

By several people, Hersholt's Andersen-translation for the English languaged world is rated as the standard translation, being one of the best.

See the alfabetical survey of fairy tales translated by Hersholt here.