TSA does not have specific size dimensions for luggage as long as it fits through their x-ray machines for screening. If the bag does not fit into TSA's x-ray machine, then the bag has to be checked
Individual airlines do have size restrictions on carry-on bags, and weight restrictions for checked luggage and vary depending on the airline. TSA recommends that passengers find out individual carriers size restrictions on bags to avoid issues at the gate after passing through TSA. For example, Delta and United Airlines allow a small personal item plus one carry-on bag that is approximately 22 x 14 x 9 inches. The carry on items must be able to fit into the overhead bins or has to be checked and picked up at baggage claim. Spirit Airlines allows bags that are 22 x 18 x10 inches for carry-on luggage.
Some airlines make exceptions for small musical instruments as long as they can be accommodated in a closet or in an overhead bin inside the airplane's compartment. TSA is responsible for screening all baggage before it enters the secure area of the airport, however, they are not responsible for ensuring that bags meet the individual requirements of each airline.