With ample storage and a fit designed for easy casting, vests are the standard when it comes to fly fishing. A vest with a good ratio of small pockets to large pockets makes it easier to organize your gear. Consider a vest with mesh ventilation panels if you fish in particularly hot weather.
Chest and hip packs are the smallest storage options. You'll have to pack wisely, but what you lose in storage capacity you gain in mobility and quick access to small essentials. Stash one of these packs in a larger pack give yourself the option of leaving your larger bag on shore when you wade in.
Slings go over the shoulder and offer more storage than chest or hip packs but slightly less than backpacks. Backpacks offer the most storage and thus, these packs are primarily useful for overnight trips or long approaches to your favorite fishing spots.