Early South Carolina Newspaper Database
The numbers in parentheses, following the ship names, indicate the number of times the vessel is mentioned in the ESCN Database.
Early South Carolina Newspapers Database is strange beast. It is a database of entries from eight 18th century Charleston newspapers. The database will direct you to mentions of individuals, ships, slave references, advertisements, and organizations in specific issues of these eight different Charleston newspapers. In that sense, it's incredibly valuable.
However, getting the content itself is not as easy. The information listed here shows which ships are mentioned in the ESCN database, though not where. If you find a vessel name that interests you, you have two paths to get the information you seek. First, you can submit a request to the owners of the database, along with the related fees, and they'll send you what the database finds. You can find a lot of information about what to expect on their website, though it is a bit of trick to locate it all. An example vessel report appears here and the order instructions are on a variety of pages linked from here.
Or second, you can search the print volumes, one at a time, through the ship names section, in a library near you. You'll need to search every volume, because there's no way to know where to find the ship name you're seeking. Only then will you find the newspaper title and publication date. Of course, you need to be near a library that holds all of the available volumes, so it might actually be cheaper and quicker to have the database owners run the report for you, for the relatively low costs they charge.
In either case, you'll only get references to South Carolina newspapers. You won't get the content from the newspapers themselves, so you'll still need to track down microfilms of the specific SC newspapers to get the actual mentions of the vessels you seek.