There is some dispute over who sang it first with rival claims but I think Liverpool probably have the best claim. They were singing it on the terraces there shortly after Gerry & The Pacemakers released it in late 63 early 64. Celtic were also singing it from a very early date though. I spoke to one Liverpudlian years ago who said he went to the second leg of the European Cup Winners Cup semi final in 1966. Liverpool had lost the first leg in Glasgow and when he got into Anfield he was astonished to see it packed full with Celtic fans all singing "You'll Never Walk Alone" so they were apparently already singing it by the mid 60s too. Great anthem for both clubs. Here is Gerry leading both sets of fans in 2003 https://www.youtube.com/watch?v=NzJqv_Oj_W0