Finding unmarried ancestors is by no means unusual in genealogy, but finding the name of the father in such cases is not so easy. A birth certificate or baptismal entry will usually only record the mother's name, with no reference to the father. After 1875 a father could be included on the birth certificate of a child of unmarried parents, but only if he attended the registration as well as the mother.
But there can be clues to parentage. Whilst birth certificates usually don't record fathers where the mothers were unmarried, this was not always the case at baptism. For example, where a child was baptised in a small, rural parish, it is possible that the father was widely known. In which case, the clergyman recording the baptism might insert their name, with the term "reported father" or similar.
But even then birth certificates can sometimes help. Particularly for male children, the mother may give the father's full name as middle names for the child. For example, one Charles Llewellyn Evans Morris was born in Penrith in 1844, the son of Sarah Morris, who was unmarried. It would seem more than likely that Llewellyn Evans represented Charles' father and in this case it would be certainly worth seeing whether there was anyone living locally of that name.
Equally, people sometimes told the truth at marriage: a father with a surname different to the party marrying gives a clue that such a scenario happened. It might even be that the father had a role in the child's life, but for whatever reason the parents simply did not, or could not, marry. Thus it might be that the clues to parentage are there in the records all along!

