Is there a way to find what types of aircraft have landed at an airport?
Speaking for the US, airports only report total movements (landings/takeoffs) and this information is available to the public via the FAA. This information is in aggregate and does not contain the type of aircraft.

Flightaware tracks only the type of aircrafts that land and takeoff at an airport; it also tracks the identifier used. It has very good coverage globally, although some areas have better (more accurate) information than others.

For most GA flights, the identifier is sometimes also the tail number, which can then be used to narrow down on a specific airframe.

Just add the ICAO identifier for your aerodome at the end of this URL: http://flightaware.com/live/airport/ to see stats of a particular aiport; for your airport its http://flightaware.com/live/airport/KCSG