The state of Delaware has two free genealogy resources available online in the public finding aids section. The first is the Bastardy Bonds Database. The second is the Probate Records Database.
When children were born out of wedlock in Delaware, the father was required to file a bastardy bond. This was to ensure the child's welfare was provided for so they would not become a burden on the county. The bonds were administered by trustees. These bonds can yield a wealth of information including parents' names, date of birth and sex of the child, names of sureties and witnesses, amount and conditions of the bond and residence information for the mother. In most cases, the child's name is not given.
The Bastardy Bonds Database can be searched by father, mother or child's name. It doesn't appear that you need the full name. I was able to search by surname only. The initial results screen will give you the father's name, mother's name, child's name (if available), child's date of birth and county. The county is abbreviated with SC indicating Sussex County and KC indicating Kent County.
To the right of the result is a details hyperlink. Clicking on it will take you to another screen which lists what folder, container and series the bond is stored in. It also gives the parents' names, information on the child, and comments/notes.
The Probate Records Database covers all three Delaware counties from around 1680 to circa 1925. This free genealogy resource searches the index of deceased individuals for whom the file was originally created. You can search by last name, first name, date, race and county, or any combination of these. The race option is only available for Kent and Sussex Counties.
You can use a partial first name, as short as a single letter, if you have input a full last name. If you are not sure of the spelling of the last name, you can put the first letter of it and leave the first name box blank to search all surnames beginning with that letter. In the year box, you can search a year or year range. Keep in mind that the database does require the full four digit year.
The results screen will give you an alternate spelling for the surname to check and then list all the records that match your search inputs. It will list the name, date, race (if available) and county.
If you find your ancestor in either of these databases, you can request the records from the archives. Information on requesting records can be found here. Scroll all the way to the bottom of the page to download a request form. It doesn't appear that there is a search fee for requests. However, there is a copy fee.