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 Mic(hael) Carroll B 1832 (exact), St. Mary's RC parish, Co. Limerick, with the father's forename unknown, I found 1 rcd. Father's forename was John.
For John Carroll M 1831 +/- 1 yr, St. Mary's, I found 0 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.