The OP unknowingly points out why it is difficult if not impossible to answer the question. Slapping the stigma of illegitimacy on a child is taboo in modern society; so much so that we don’t collect the data in any meaningful form anymore.
In my state, if a married woman gives birth to a child, her husband is automatically listed as the father on the birth certificate. Even if husband hasn’t been in the country for the past year, he is legally the father until he goes to court to prove otherwise.
If an unmarried woman gives birth to the child, she has the option of leaving the father’s name blank, or putting any other name in that field that she desires. She can put Happy McFalkerstain in that field and nobody will question her about it. However, say she lies and says that a real person, her neighbor John Smith is the father. This ministerial act of putting John Smith’s name on the birth certificate creates no parental rights or duties in John Smith.
There is a separate action to establish paternity that must be done. So for good or ill, we have allowed the birth registries to become a farce in some cases so as not to stigmatize illegitimacy.