The earliest known MCFC specific song I've tracked down is 'Who Said City Couldn't Play". It was sung in 1930s (film of fans singing it at 1934 homecoming exists) and lasted into 1940s.
Pre-1930s fans sang local folk songs or popular music hall songs like Lassie From Lancashire. They also sang along with the band to whatever tunes they played.
1950s the Bless 'Em All adapted song was created for the 50s team.
Of songs still being sung: Never Felt More like Singing The Blues goes back to late 50s/early 60s (my guess is that it was sung at some of the derbies in mid-50s but older fans may know).
'We've got an M an A.... we come from Maine Road, Manchester...." - goes back about 60 years too.
Of course, "You'll Never Walk Alone" was definitely being sung by City fans in the 1960s (and 70s) too!