Your geographical location can make your harness storage choices for you. Up north here, we want our leathergoods kept inside over winter. Doesn't let them freeze.
Leather is made of cells, and good conditioning makes leather cells full and flexible. If the cells freeze, they burst the tissues, and leather gets limp and floppy. We keep our harness, saddles, leather stuff in a heated tack room over the winter. We have LOTS of harness, drive Multiples mostly, so storing it well is quite important year around.
Those in the southern states have said how fast mold will grow, so they have dehumidifiers going in insulated tack rooms or closets, for leather storage.
We are kind of in line with Thomas, using harness racks for most storage in the tack room. Good harness racks keep things shaped as they are designed, not creasing or bending parts. However in the summer, with our humidity so high, we keep the daily use, synthetic harness out in the open barn, so breezes blow and remove dampness. It is stored on a jumping pole hung from the rafters. Pole width is just the right size for the harness saddles and bridles. Supports and spreads out the weight of harness hanging down. Lets the harness dry easily, not in the sun. We can get all four horse harnesses on the pole and spread to dry out.
Size and diameter of object you hang bridle or collar on is real important, so you get no cracking or folds, like hanging them on a nail would do. Folks have said that hose storage hangers are a good radius for traces, saddles, reins, and prevent creasing or fold lines.
Unused leather harness here is stored mostly in good tack trunks in the tack room. Spread out, with towels between layers. Not piled too deep, so leather is crushed under the weight above. Buckles unfastened to prevent creasing. Leather strap goods coiled for storage.
Going to shows we pack things in rolling suitcases, or duffle bags. Bridles in bridle bags that help keep shaping of browbands, blinkers and cavesons from crushing. With carriage harness for our large horses, a number of the harness bags are not heavy enough to manage them. Plus storage enroute, may be crowded, so the suitcases work pretty well for us. We have places to hang harness in the trailer once horses are unloaded.
Harness bags are very popular with people who use light harness. Saddlebreds, fancy stepping ponies, QH, with the thin leather strapping, often no breeching. They would probably be using the very light carts behind such horses. Light or Fine Harness is the style needed in their breed showing.
So lots of choices, all depends on kinds of harness, where you are, and what you plan to use your horse to do in Driving activities.