When a woman who had no means of support gave birth to an illegitimate child, the woman often appeared before the county court to name the father and obtain support for the child. The goal was to prevent the child from becoming a charge upon the county.
Most, if not all, counties have bastardy case files. They are usually located among loose county court papers and may also be recorded in the county court order books (court minutes). Bastardy case files usually contain the following information: name of the mother, name of the alleged father, sex of the child and the age at the time the case was filed in court. Keep in mind, however, that if the woman and/or her family preferred to support the child, there was no need to take the case to court and no bastardy charges were filed.
The bastardy cases for Caldwell County are filed in archival boxes in the county clerk's office in Princeton, Kentucky. The files are dated 1815 - 1924, but I found one dated 1812. The following cases are among the earliest filed in Caldwell County.
Polly Newman charged Hiram Pinnel as the father of her child, who was born 20 July 1815. Summoned to testify were Joseph Easley, James Clayton and Old Mrs. Newman to appear in Eddyville 17 December 1816.
Polly Bird charged Milton Finley as the father of her female bastard child, who was delivered 7 November 1817 at the house of Thos. Hancock.
Polly Guess, single, swore on 15 October 1819 at the house of Joseph Guess being on the road leading from Princeton to Flins Ferry about 12 miles from Princeton that she was delivered of a female bastard child and the child is likely to become chargeable to the county and has charged Robert Hooker, laborer, with being the father. Hooker posted bond on 18 October 1821.
Mary Freeman, single and not a slave, delivered a male child 24 April last  at the house of Daniel Warmarlsdorf in Livingston County and Preston Grace is the father of the child. Mary is unable to support sd. child herself and prays for relief.
Priscilla Armstrong, single woman, was delivered of a female bastard child, who was delivered 28 December 1820, and is likely to become chargeable to the county and hath charged Isaac Sampson, laborer, of having begotten her with child. The charge is dated 29 December 1820.
Abigail Johnson was delivered of a female bastard child 26 January 1822 at the house of Benjamin Johnson and charges James M. Hall as the child's father.
Matilda R. Duning was delivered of a female bastard child 4 February last  and charges William Galoway with being the father.
Polly Smart swore John Bolton, yeoman, was the father of her male bastard child born 20 May last .