According to the Transportation Security Administration (TSA) and most airline regulations, a shuttlecock is generally considered a small item similar to a Ping-Pong ball or a tennis ball. As such, you are allowed to bring it in your carry-on bag or checked luggage.
However, if you have any concerns about taking your portable badminton equipment on board an airplane, please consult with the airline’s customer service or check their official website for specific guidelines and restrictions on carrying sporting goods as cabin baggage.