Hi Steve
Inconsistency in spelling of surnames is well known to those who have conducted research into their Irish family history! You will find that in the context of Irish historical records there are many spelling variations of the same name which means that in conducting any research you have to be aware of potential variants. An excellent starting point for surname research is the ‘Surname Search’ option at
https://www.johngrenham.com/surnames where you can explore the location, frequency, variant spellings and history of the McMullen surname.
Prior to 1864 and civil registration of births, deaths and Roman Catholic marriages (1845 for non-Catholic marriages) in Ireland you will have to rely on church baptism, marriage and burial registers to confirm ancestry details. Church registers are effectively the building blocks of Irish family history back beyond the mid-19th century.
In the ideal world family history databases such as Ancestry or Rootsireland.ie would record a baptism of James Herbert McMullin born c. 1830 and the marriage of his parents, William McMullen and Catherine; or a ‘relative’ in the wider family history community (perhaps identified through DNA) holds information/oral traditions, documented and passed down through the generations, that sheds new light on ancestral origins.
RootsIreland.ie is the largest online source of Irish church register transcripts, but it must be emphasised that a failure to find relevant birth/marriage entries in this database (or indeed any of ‘big’ family history databases) doesn't mean that the events you are looking for didn't happen in Ireland. It simply means that they are not recorded in the database; for example, they may be recorded in a church whose registers don’t exist/survive for the time period of interest or in a source that has not been computerised.
Furthermore, 'census substitutes', such as early-19th century tithe books or mid-19th century Griffith's Valuation, by naming heads of household only, provide insufficient information to confirm the nature of linkages between named McMullen households in these sources. Census substitutes, however, are very useful in confirming the presence of a family name in a particular townland and/or parish, and in providing some insight into the frequency and distribution of the McMullen surname in Ireland.