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“Carrot on a stick” or “carrot and stick”?

There have been several debates around whether the idiom is “carrot on a stick” or “carrot and stick”.  Nobody can say which one is more accurate than the other, since explanations from both are reasonable.  However, “carrot and stick” seems to be more commonly used by many.

There are also arguments around that the phrase should be “carrot or stick”

See  

Carrot and stick

http://en.wikipedia.org/wiki/Carrot_and_stick

Carrot on a stick

http://en.wikipedia.org/wiki/Carrot_on_a_stick

Carrot on a Stick vs. Carrot and Stick

http://www.boston.com/news/globe/ideas/articles/2004/05/02/scuff_links/

http://www.phrases.org.uk/bulletin_board/3/messages/236.html

Carrot on a stick vs. Carrot or stick

http://www.wsu.edu/~brians/errors/carrot.html

Carrot or stick

http://www.encyclopedia.com/doc/1S1-9199311290337893.html

Carrot on a stick

Carrot and stick

Carrot or stick

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