onion I just plant the root part of the onion in moist loose soil.

Chere

a. can broccoli be grown from a bit of stalk (how little)? No. Broccoli grown in the garden and left unpicked soon will flower (what you buy in the store and we eat are the flower buds) and seeds will, of course, follow. The stalks themselves have no regenerative ability.

b. belgian endives from their stalks( since lettuce and cabbage work from their stalk)? No. You need roots. A Belgian endive, or witloof chicory, is planted from seed and allowed to grow into a plant all summer. The roots are harvested in the fall before hard freezing occurs, the foliage removed and the roots are stacked in the field or other cold storage area. After they are exposed to cold the roots are planted upright in moist sand and forced to grow a new head by keeping the air temperature near 64 degrees.
c. kale, collard, mustard greens that don't end in just a leaf tip, but are still connected, thus have about a 1/2 inch to an inch of stalk --would that grow? No. There has to be some root attached, and even then it would be iffy. Plus, if it did grow, the plant would bolt to seed quickly.
d. how about Brussels sprouts bought on their stalk, would the stalk grow brussels if planted? Again�NO. If you had grown the stalk in your garden one year, the plant would (if it survived the winter) then produce a flower stalk that would provide seeds for you to plant.
e. will a turnip or rutabega root grow if you plant the greens with 1/2 inch of stalk, or even if you just plant that course stalk with no greens? Not from a stalk alone, but if you were to slice off the crown (the top of one of these that still has the greens, or at least a little of the green stalk, attached) along with a half-inch of the root, you might get a plant to grow from that. But that would then provide you with seeds since it would send up a flower stalk in the garden�you would not get a new edible tuber from that plant. I got my start of horseradish by choosing a crown from the grocery store (a section that had not been trimmed to closely to the root) that showed some tiny sprouts. I kept it in my refrigerator, wrapped loosely in paper towel and in a baggie until early spring when the ground had thawed enough for me to plant it. Horseradish also will grow from any of the side roots left in the ground or planted.
By the way�I almost never have seen turnips or rutabagas that have not had the crown cut off entirely�and rutabagas are always waxed to keep them from dehydrating and getting soft in storage.
Carrots, parsnips, and probably some other root vegetables will re-grow as described for horseradish�however, these plants will not form the long edible roots we use but, as with turnips, this would simply provide you with a flowering plant from which you would get seed. (Which may or may not be viable if the original plant was a hybrid!)
Jerusalem artichokes (totally misnamed since they are not related to artichokes nor have they anything to do with Jerusalem), often marketed as "sunchokes" may be planted as is (the tubers may be left whole or cut into "eyes� like white potatoes) and you will get a tall plant with yellow daisy-like flowers (it actually is a member of the sunflower family) and in the winter or early next spring you can dig down & harvest a LOT of lovely tubers. Very good eating fresh (a lot like water chestnuts) and especially good for diabetics as the starch does not convert to sugar. I do not care for them cooked as they do not have the same texture as white potatoes.
As Chere mentioned in her post, one may grow scallions (baby onions) from old onions that may have softened and are sprouting in the bottom of your veggie bin. If you do not then pull the young onion to eat, left to grow it will flower & provide seeds. (You will not necessarily have onion seeds that will be successful in your growing zone, though. Safer to order from a catalog�and planting plantsyou can order is much easier than seed!)
Sweet potato plants ("slips") may be started in late winter to get little plants for the garden�IF you can find one that will sprout since most store sweet potatoes have been treated with growth retardant. I had to sacrifice a number of potatoes before I finally found one that would sprout shoots from the tuber. There are direction posted in Gardenweb at http://forums.gardenweb.com/forums/load/market/msg0319531224783.html
For amusement, one may start most any citrus from seeds but consider them house plants. Same with avocado (start a pit the way you�d start a sweet potato).
What you need is to get a good, elementary botany book and another on plant propagation from your local library (or buy them used from amazon) so you may learn the basics of plant reproduction and propagation.
Finally, remember when you post in Gardenweb to add your growing zone and general geographical area to your signature line so others may help you better. (Like mine�central OH5)