Flying with a suit with American Airlines
American Airlines allows passengers to take a soft-sided garment bag measuring up to 51 inches / 130 cm when combining length + width + height. This item can be taken as your carry-on item.
Flying with a suit with Delta
Delta do not specify rules about transporting a suit in a garment or suit bag, but their carry-on allowance is one carry-on bag and one personal item free of charge. The main carry-on bag must fit the combined dimensions (length + width + height) of 45 linear inches or 114 cm., and must not exceed the dimensions of 22 x 14 x 9 inches. Therefore, you can carry your suit on board as long as it fits the dimensions permitted for carry-on luggage.
Flying with a suit with Southwest
Southwest Airlines will allow garment bags as a personal carry-on item, as long as it fits the dimensions of 10 x 16 x 24 inches. Although many garment bags are larger than this, as they are soft-sided and can be folded, they will fit the given dimensions and therefore will be accepted.
Flying with a suit with United Airlines
As with Southwest Airlines, United Airlines will generally allow suit bags as the carry-on bag, as long as it fits, when folded or otherwise, as a carry-on bag. The permitted maximum dimensions of the carry-on bag are 9 x 14 x 22 inches.