The pocket square has a billion possibilities. Generally, my rule is that the pocket square can be made out of nearly any fabric, color, pattern, or thickness. Usually I build it together starting with these rules. First, the presentation: In most cases, the pocket square shouldn't match the tie exactly, though certain exceptions are allowable. The boldness of the pattern will also dictate the level of presentation... bright and busy gets a low profile out of the pocket, while plain or subdued can be grown out a little. The pocket square can be similar in color, pattern, or material to the tie, and to whatever degree one dares.
Note: when in doubt, use white.
Now for the fold: The fold is generally dictated by the pattern and material. Many thin silk pocket squares will slip down into the pocket over and over, and require a special type of fold, or a tight square folded plain handkerchief at the bottom of the pocket to act as a wad, while the visible one sits ontop of it.
There are several folds: the Fred Astaire (puff-fold), is good way to deal with a heavily patterned pocket square.
Too many peaks and blossoms from a busy patterned material is far too flash, and it creates parody. A plain or subdued version gets the flared method (some call this the tulip). The 3 peaks method is great for subdued styles as well, but I find the 4 peaks a bit too much.
Always avoid the dreaded and terrible "labial" style (pictured in yellow below, with identical tie).
I have developed several folding methods that compensate for under-sized cloth, and one method for very large pocket squares as well. In terms of where to get them, any clothing department will have them. I go to fabric stores and grab silk, cotton, etc, and make them myself, I buy entire lots of them on Ebay, I have even used the back-panels of old decommissioned shirts.
As for angles, I prefer sweeping away from my centerline, and at an angle that compliments my collar and lapels. Mostly, it should not look carefully done, and should be pulled out at a moments notice for the appropriate situation. Here is a simple presentation:
It's clean and easy, won't draw attention, and it sweeps away at just the right angle, emulating the opposite lapel notch. Next, We have William Powell:
His pocket square is reaching right up to the outer boundary of display, just before it becomes overflown. Cary Grant's pocket squares often angled towards his centerline, and up... probably to draw the eye towards his face, which seems like wasted effort, as all eyes reported directly there anyway:
Jackets just tend to look empty without them. Business suites need them, hunting tweeds should have them (tending towards actual use: wiping face, gun barrels, the top of a bottle). Another never: the fake. These are press-folded cotton peaks stapled to cardboard. It looks awful, but imagine the embarrassment when you take your seat on a plane, and the young lady next to you is sobbing because her visit with her hometown beau has come to an end, and she is headed back to college. Will you hand her the cotton and cardboard 3x5 in your pocket? Unforgivable.
***Always carry another plain cotton hanky for this reason, and if you do give your pocket square to one who needs it, expect that she will keep it.
Groucho Marx used to achieve four peaks by using a white glove. It's silly, of course, but I like it. A certain club in Boston hosts a Seafood Night for its members, and all are encouraged to wear a flat seashell in their pocket. Botany Night usually sees members wearing small series of green leaves instead of cotton or silk. I have never stayed long enough to see what was used during Lectures on the History of Surgery.
In the image below, a self-described "Dandy" produces 3 peaks that seem to outgrow their vase:
My tolerances generally require restraint with the pocket square, but others may encourage blooming of this sort.
I got an email asking which method of folding is correct. The answer: none are incorrect.
Generally, if the jacket has a breast pocket, you should always have something in it. Many coats (the Chesterfield, for example) also have pockets, and a pocketsquare is correct there as well, though you should keep it simple, like polka dot or dark patterns. Ensure that your pockets are opened, if the jacket is ready-made by carefully removing the stitching with a tiny pair of snips, and then unthreading the loops.
Try out different patterns, but remember that if the pocket square is too precious to hand to somebody, either don't wear it, or carry a back-up.
From New York:
I just wanted to back Boston up on this, and emphasize the importance of two things he mentions: that the square is meant for use (ideally, to dry the tears of a lady) and naturally ought not be replaced to the pocket once it has been soiled, and secondly that it should never be too deliberate-seeming. Certainly, deliberate about them in general as a gentleman should for everything he does, but do so in advance so that the actual wearing can be unselfconscious.
I'll also add one opinion on a more general level. It seems to me that it is better to wear a pocket square without a tie than a tie without a pocket square, except in the aforementioned case of a damozel having just emptied her sinuses therein, in which event you could switch to a clown suit with gentlemanliness unquestioned. In the same way, I would say that it is better to wear a jacket with no tie than a tie with no jacket, unless one is a bouncer, a barber, or a barman. This may, however, be a matter of personal preference; I leave it to the floor.