Hang it handle down in belt
Don't use a metal scabbard, else you will look like a rookie or Montana weed cutter.
Just slide the handle down through your falling belt (supported by harness for best results). The belt will secure the ax. Your wedge pouch (leather) will hold two to three wedges.
I would typically keep my falling ax with my gas & oil jugs. For wedges, you should have two plastic on one long aluminum wedge. Use your aluminum wedge as your driver, that is, hold the wedge like a club and drive the smaller wedges in. It works great for most trees. Believe me, I have cut the big wood in the rain forest and was a top-producing cutter.
It is funny to listen to the advice given in this thread, so I registered and wanted to give you the best advice.
Good luck and don't use the metal scabbard, else you will look silly with this needless attire.