Shallots and onions:
Cut of about 1cm of the bulb from the bottom, where the roots grow out, and press them into moist soil. Eat the rest.
For larger onions, you can find multiple centres and cut them apart to further divide them.
Cut off the bottom part, which bulges slightly, and again, bury that in moist soil; clean off and eat the rest. Leaves should grow quickly. Leeks need more space and will choke out their neighbours, so move them to pots of their own, or anywhere relatively sunny if the ground isn't frozen. The same for scallions, but I don't think they are as aggressive.
It is possible to do the same as with onions and cut a portion off the bottom, and this is sometimes successful. I have found another way, which only works with older garlic that has fully developed leaves in the core. That method is to peel off the white fleshy outer leaf for eating, and plant the central meristem portion, again in moist soil. They should begin rooting immediately, and growing new leaves within a week or so.