Variation of "cut a rug" (i.e. 'to dance enthusiastically') - a term that became popular jargon in the 1920s.
The basis for the phrase comes from wear created by over-use of (by dancing on) an area of carpet.
Used in The Blutones' song 'Cut Some Rug' (You say I can talk to you anytime, Though I just wanna cut some rug).
This is a great song! Wanna get up and cut some rug?
87👍 13👎