De esta página tan interesante → http://www.rgagnon.com/javadetails/java-0487.html he encontrado la manera de obtener el mimetype de un fichero dado. Aquí va el código de ejemplo:

import javax.activation.MimetypesFileTypeMap;
import java.io.File;
 
class GetMimeType {
  public static void main(String args[]) {
    File f = new File("gumby.gif");
    System.out.println("Mime Type of " + f.getName() + " is " +
                         new MimetypesFileTypeMap().getContentType(f));
    // expected output :
    // "Mime Type of gumby.gif is image/gif"
  }
}

Dice en la página:

The MimetypesFileTypeMap looks in various places in the user's system for MIME types file entries. When requests are made to search for MIME types in the MimetypesFileTypeMap, it searches MIME types files in the following order:

- Programmatically added entries to the MimetypesFileTypeMap instance.
- The file .mime.types in the user's home directory.
- The file <java.home>/lib/mime.types.
- The file or resources named META-INF/mime.types.
- The file or resource named META-INF/mimetypes.default (usually found only in the activation.jar file).