Curlicue means: “A fancy twist or curl, such as a flourish made with a pen.”. Curve is the essential means in decoration of all human artifacts.

Filigree means: “Delicate and intricate ornamental work made from gold, silver, or other fine twisted wire.”.

A iron wrought curlicue door of a shop in The Dessert Passage shopping mall in Aladdin Hotel-Casino at Las Vegas, USA, 2003-10.
brighton curlicure
A door of shop Brighton Collectibles at Westfield Valley Fair shopping mall in San Jose, CA, USA, 2002-07.
curlicue mat
A doormat with curlicue pattern. Photo taken in Mountain View, CA, USA, 2004-12.
cathedral Duomo di Milano
A facade of a cathedral the Duomo di Milano in Milan, Italy. It is not curlicues exactly, but shows the intricate use of curves. (Photo taken in 2005, courtesy of Paula Fonseca)
Rathaus Vienna City Hall
Arches of City Hall of Vienna, Austria. Photo taken in 2005-10 by Sol Kawage, used with permission.

[etymology of curlicue]

1843, American English, from curly + cue, perhaps from French queue “tail” or letter Q in its looping script form. Earlier in this sense was curlie-wurlie (1772).