I travel light: Small bi-fold from mont-bell, nylon mesh so that it dries easily when I sweat (doesn't stain like leather). Thinnest I've found for a back pocket, no metal snaps/zippers/emblems that would be lumpy.Your wallet ... is "a picture of your life." It especially reflects how you think about money and manage your finances.
Not much in it. Driver's license, gaijin card, copy of health insurance card, a car insurance contact card (number to call in an accident), copies of my My Number card (to have the number on hand) and of my 傷害者手帳 (too fat to carry everywhere). Points cards for two stores (little used).
I only carry money in it when cycling--bills/change then go in it, and then that goes in a waist pouch. (No pockets in cycling shorts, some jerseys do have them.) Other than this, money typically goes in a regular pocket.
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How about you?