Moving Here Migration Histories website. The top ten postcodes are:
1. M14 (2,352)
2. M19 (2,122)
3. M20 (1,972)
4. M32 (1,546)
5. M33 (1,505)
6. M16 (1,454)
7. M25 (1,432)
8. M10 (1,399)
9. M21 (1,325)
10. M13 (1,322)
M14 is Fallowfield, Moss Side, Rusholme, Ladybarn; M19 is Levenshulme, Burnage, Heaton Mersey, Heaton Chapel, Reddish; M20 is Didsbury, Withington; M32 is Stretford; M33 is Sale, Brooklands; M16 is Firswood, Old Trafford, Whalley Range.
So the top six postcodes are all clustered around south Manchester.
M25 (where I grew up) is Prestwich, Sedgley Park, Simister. What used to be M10 (now M40) is Collyhurst, Miles Platting, Moston, New Moston, Newton Heath. These two postcodes are north Manchester.
M21 is Chorlton-cum-Hardy, Barlow Moor; M13 (as Ian Brown famously sang) is Ardwick, Longsight, Chorlton-on-Medlock. Both postcodes are in south Manchester.
This clearly shows that the Irish settled, at least in recent years, mainly in south Manchester, around the two football stadiums, Old Trafford and Maine Road, and the university. It follows that despite a widespread perception of Manchester United as a 'Catholic' club (especially from the 1930s to the 1960s), lots of Manchester Irish would support City, members of the band Oasis being the classic example (although we shouldn't forget Johnny Marr from the Smiths).
It also suggests that the north and south of Manchester are the immigrant areas. The north is actually more of an immigrant area than is suggested in the information above. The Irish World Heritage Centre is located in Cheetham Hill, probably the area of Manchester most defined by various waves of immigration.