1st of February and St, Brigid is apparently very angry. Just off the top of my head and without googling or searching on I Tunes, I tried to come up with 12 songs that mention one of the months of the year.
January - January by Pilot
February - American Pie by Don McClean
March - Tramp, tramp, tramp, the Boys are marching (?)
April - Pride by U2
May - Come what may - Vicki Leandros
June - The Blue Ridge Mountains of Virginia - Laurel and Hardy
July - Not sure who sings it, but that irritating song that goes mamamamamamammamamaamamaamamamaJuly
September - September morn by Neil Diamond
October - U2 again?
November - something about "I'll be back in November"
December - Oh what a night - Franki Valli and the Four Seasons