Tailor's Note: April 2021 - Which pocket style is for you
Disclaimer: These ideas are based solely on my opinion, please feel free to ignore, flout and disobey at every opportunity.
Interestingly, this is the original pocket. Before the humble patch, men (and women but less commonly) held their possessions together in a small purse on a belt. Think 16th Century fanny packs. But thieves found them remarkably easy to pinch so people began to start hiding their valuables closer to their bodies, asking clothes makers to sew their purses directly onto their clothes. And thus the patch pocket was born.
As a result of the simplicity required to install a patch pocket, it is considered the least formal. It is a really good option for someone looking for versatility in their tailoring. A coat with patch pockets can easily be worn with jeans or chinos as well as part of a full suit. Plus patch pockets are roomy and practical.
Straight with a flap
This is the option seen most often, particularly on RTW suits. It is particularly useful for people who use their outside pockets because over time heavy use can cause pockets to sag a little. The flap will hide that wear and tear, keeping a coat looking smarter for longer. Although this pocket can look really smart, it is also very practical - you can throw in a wallet, keys, a phone etc (disclaimer: heavy items may affect the line of the jacket after time) so it is appropriate for both a formal work suit and more casual looks.
Slanted with a flap
This slanted variation was invented for hacking jackets. The slanted angle made it easier to get into whilst on a horse. There is a theory that the angle draws the eye up towards the waist which is the narrowest part of you, and therefore is more flattering than the straight pockets. I'm not sure I buy that, but it can be a really nice alternative to the straight pockets.
I never encourage this type of pocket unless you don't use them. This type of pocket should probably be sewn up to discourage heavy items being thrown in and ruining the shape of them. As such I would only really put them in very formal garments like a Dinner Jacket. The jetted pocket is very discreet which is great for something streamlined and probably quite form fitting like a Tux.