These are the songs that reached number one on the Top 40 Best Sellers chart (expanded to the Top 50 as of October 13) in 1951 as published by Cash Box magazine. Artists were not specified in the charts of this period so songs may represent more than one version. The artist who most popularized each song is listed.
Issue date Song ArtistJanuary 6"Tennessee Waltz"Patti PageJanuary 13January 20January 27February 3February 10"My Heart Cries For You"Guy MitchellFebruary 17February 24March 3"Be My Love"Mario LanzaMarch 10March 17March 24March 31April 7"If"Perry ComoApril 14April 21"Mockin' Bird Hill"Patti PageApril 28May 5May 12May 19"On Top of Old Smoky"The WeaversMay 26"Mockin' Bird Hill"Patti PageJune 2"On Top of Old Smoky"The WeaversJune 9"Too Young"Nat King ColeJune 16"How High The Moon"Les Paul and Mary FordJune 23June 30"Too Young"Nat King ColeJuly 7July 14July 21July 28August 4August 11"Come On-a My House"Rosemary ClooneyAugust 18August 25"Because of You"Tony BennettSeptember 1"Sweet Violets"Dinah ShoreSeptember 8"Come On-a My House"Rosemary ClooneySeptember 15"Because of You"Tony BennettSeptember 22September 29October 6October 13October 20October 27November 3"(It's No) Sin"Eddy HowardNovember 10November 17November 24December 1December 8December 15"Down Yonder"Del WoodDecember 22"(It's No) Sin"Eddy HowardDecember 29
See also 1951 in music List of number-one singles of 1951 (U.S.)
References https://web.archive.org/web/20110515183302/http://cashboxmagazine.com/archives/50s_files/1951.html