Top hats should be worn on the front of the head; when indoors, they are carried under the arm.
The wearer should know how to 'doff' their hat - in other words, raise it above the head to greet guests.
Black silk is the ultimate choice, but such hats have not been made since the 1950s. The purchaser of a second hand model should be prepared to pay handsomely. Grey felt is the alternative, and is now the norm.