A white pocket handkerchief, like any element of a man's attire, is most elegant when it appears to be arranged nonchalently, casually. But that doesn't rule out a square fold.
A friend told me recently how he always stuffs in his white (linen) handkerchief points uppermost, in the morning and whever forced to use it. He folds it in half, stuffs in the middle and lets the points fall as they please. It is a relxed, degagé look.
But you can stuff a square fold too. People always assume that a square fold, where the handkerchief edges run across the top of the pocket rather than sticking out of it, has to be millimetre-perfect and razor sharp. Why?
Sure, that's how the arrangement was popularised, by TV presenters back in the 1950s and more recently by TV shows like Mad Men. But that isn't the only way to do it. Besides, a fold that sharp is so tough to achieve and then maintain. The geeky effort involved is there for everyone to see, on display.
Instead, I recommend just folding the handkerchief twice, or enough such that the width is less than that of the pocket, and stuffing it in square. It's something I see my friends at Kiton do all the time, and something I admire. You can see the effect in my two recent photos - of the A&S suit and the wedding shot too.
The corners do not remain perfectly aligned. They are caught by the friction of the pocket and wriggle into a random arrangement. As with a 'points' approach, you can tug at the corners as much as you want if you judge the look needs a little adjustment. Just don't make it look too perfect. Oh, and stuff it so the points are to the outside and at the front of the pocket.
Points can look just as studied as a square. And to my taste, points will always be a little dramatic.