Combine the corn flour, all-purpose flour, seafood seasoning, salt and pepper in a large bowl. Whisk together the milk and egg in a small bowl and season with salt and pepper.
Add enough oil to a heavy-bottomed pot fitted with a deep-frying thermometer to reach 4 inches up the sides. Heat the oil to 365 degrees F and line a sheet tray with paper towels.
Dip the clams in the milk, then into the seasoned flour. Add 8 to 10 clams to the oil at a time and fry in batches until they are crispy and golden brown, 45 seconds to 1 minute. Remove the clams to the lined sheet tray. Continue frying the remaining clams.
Sprinkle the clams with salt and pepper and transfer to a platter. Serve with the Tartar and Cocktail sauces and lemon wedges.
Stir together the mayonnaise, relish, dill, lemon juice, salt and pepper in a medium bowl and refrigerate at least 1 hour and up to 2 days.
Stir together the ketchup, chile paste, horseradish, lemon juice, Worcestershire, salt and pepper in a medium bowl and refrigerate until serving, or up to 1 week.