I research the Irish Family History Foundation's database. It contains 12,800,000 surviving rcds for abt 70% of Irish counties. Rcds are increasingly sparse B4 1840, rare B4 1800. Cos. Donegal and Derry were recently added to the database.
Note 1 - Without a KNOWN county, the chance of an ancestor's rcds NOT being in this database is AT LEAST 30%.
Note 2 - Only 1/2 of Co. Cork's rcds are in this database.
Note 3 - The RC church kept control over most parish rcds from Tipperary (north). These are available thru the Tipperary Family History Research center at a substantial fee (regardless of outcome), and the National Library of Ireland on microfilm.
For Br(idget) Halligan B 1869 (exact), Knock RC parish, Co. Mayo, with a father forenamed Pat(rick), I found 1 rcd.
For Marg(aret) Halligan B 1871 ditto, ditto.
For Hon(or) Halligan B 1876 ditto, ditto.
For Ellen Halligan B 1879 ditto, I found 1 rcd in 1883.
For Tho(mas) Halligan B ???? Co. Mayo, with a father forenamed Pat(rick), I found 1 civil rcd: 1881 Holymount CP.
For Pat(rick) Halligan B ditto, I found ditto, 1866.
For Mic(hael) Halligan B ditto, I found ditto, 1873.
For John Halligan B ditto, I found 5 rcds, but NONE were in either Knock RC parish, or Holymount CP.
For Mary Halligan B ????, Co. Mayo, with a father forenamed Pat(rick), I found 4 rcds within a relevant range: 1864 & 1868 civil parishes, and 1864 & 1868 church parishes.
For Pat(rick) Halligan M 1860 +/- 4 yrs, Co. Mayo, I found 3 rcds: 1862 Kilcommon & Robeen, 1857 Crossboyne & Tagheen, and 1858 Kilcommon RC parishes. One of these MAY be for this ancestor. A view of the detailed rcd SHOULD show the bride's name which can be matched with the mother's name found on the children's B rcds.
Google the Foundation's name to lead you to this database. Register, opt to be notified as rcds are added and browse. Pls post your search results to my genforum msg.