The Traditional Way
The standard way is with a garter belt (in the UK it is called a "suspender belt"; it may also be called a garter; hell, sometimes it's even mistakenly called a girdle, though girdles are actually different as they do this magical thing called shaping but also makes you look like you're wearing tragically huge granny panties). Now, here's a brief tutorial about the garter belt that is necessary for you to know. Garter belts are sort of like underwear that you put over underwear, and it's got straps with clips that hang down to attach to the top of the stocking so that it stays up. You've probably seen it in lingerie catalogs, where scantily clad women pose in these lace-bedecked, satiny confections with dynamite legs other less fortunate women only dream of.
Now, if the only place you've seen it is those said catalogs (Victoria's Secret catalogs are the poor man's porn), take a long, last, longing glance at these, and then chuck the damn catalog out the window. Those garters you saw in the catalog? They are not garter belts. They are temporary reprieves for the woman before she gets naked in a bed. They are not meant to last for an entire day walking, running, fetching coffee for your boss or kicking ass in a Crouching Tiger, Hidden Dragon way. If you are not built like those Victoria Secret models, then those lacy bits and ties will get lost in the folds of your skin and you'll be adjusting your belt just as frequently as you'd be adjusting your stockings without garter belts.
Do note that in lingerie catalogs, the garter belt is put on over the knickers, and that is how society accepts it. This, however, can be done the other way - knickers over garter belt - as the benefits are twofold: 1) the garter belt will stay up, aided by your knickers and 2) if you are trying to seduce someone to your bed, it is much quicker to take off your knickers this way.
Real garter belts are no-nonsense sorts of things. They tend to sit high on the waist (less adjusting; I've seen lower ones, to accomodate the whole "hanging on hip" clothing that the Brazilian jeans trend brought, but I've never seen a good one), and the ones that will serve its purpose well will have six or more clips hanging down on your legs; three for each leg, though I've seen as many as six for each leg, which is kind of overkill. Two for each leg is adequate (four clip belts), however, for unpatterned items; four clips for each leg (eight clip belts) are especially good if you have patterned stockings that have to stay in a certain place to look right (e.g., seamed).
Remember, folks, make sure you buy garters with metal clips. Plastic clips, which are common in cheap, low quality garter belts from cheap, low quality lingerie stores (Fredericks), tend to slide right off stockings. They are as annoying and hazardous as detachable bra straps that don't stay attached. Unless, of course, your whole plan is to wear these garter belts long enough to seduce the object of your choice into bed.
However, no matter how hard you will try to avoid it, adjustment is necessary. Run to the bathroom as often as necessary (once or twice a day), surreptiously adjust, rinse and repeat. This is especially true for anything with patterns.
Corsets can also be used to hold up your stockings (as many of them are built with hooks for garter straps), but this tends to be very strictly for bedroom and or fetish show, unless you wear a corset underneath your clothes (which is really kind of a faux pas nowadays).
The last item that will also work in a satisfactory faction are bustiers, which are corset styles without boning and are really for shaping purposes (creating a smooth line). Paired with stockings, they reduce the amount of visible bumps underneath a dress or shirt and are a good alternative to simply wearing a garter belt. They are better known as basques in the UK.
The Not So Traditional Way
Them Japanese girls are clever, I grant you that. During the height of fashion of loose socks, especially as they got baggier and baggier and longer and longer (100cm and longer!), there was the problem of keeping them up that was solved with something that only Japanese people could think of: sock glue!
Yes, it does exactly what it say. You glue your stocking (or sock) to your skin, and it stays up all day. No, I cannot say what it's like from voice of experience, but I've seen them for sale.
Can't find this mysterious sock glue that you so greatly wish to use? Try the next handiest thing, which you can find in art stores: liquid latex. You will have stockings that stay put all day with this. But peeling it off your skin is rather like peeling a band-aid - your mileage of pain may vary. And this works. Keep that in mind if you can think of, uh, any other uses for it.
Both items work better for socks of any sort, but liquid latex works in a jiffy for stockings. Just not sure how it washes off.