There are an awful lot of bulbs, and those we call bulbs, to be able to say "most" but many do. Al

Almost all bulbs are capable of multiplying through division or offsets.

However they are all adapted to their own environments and in a garden they all have differences in performance. It can really depend on not only the species of plant but on each variety, which has different abilities to overcome problems it faces in the garden. Many bulbs you buy are not well-adapted to your climate and will gradually fade away rather than multiplying.

Grape hyacinths will multiply very quickly.
Garlic cloves in a garden will multiply quickly.
Crocus will multiply quickly.
Daffodils will multiply gradually.
Ornamental alliums, depending on variety, can multiply slowly or quickly - small kinds multiply more quickly.
Dutch iris are very difficult to multiply at all.
Asiatic lilies can multiply quickly.
Tiger lilies can multiply VERY quickly.
Oriental lilies usually multiply slowly.
Gladiolus will usually divide into exactly 2 new large bulbs per year, plus a bunch of very small offsets that will take several more years to reach blooming size.

The ideal is to get a multiplciation of large, healthy bulbs, or multiplication that favors large bulbs so when the bulbs become overcrowded they naturally stop dividing. This is usually done by growing them in their ideal soil type and with sufficient water. Some kinds of bulbs that are subjected to too much drought or heat stress have a tendency to split themselves up into a large number of relatively small bulbs - too small to bloom well, and in competition with each other, and plus they divide even more the next year and eventually kill themselves off by getting too small.

That's especially common with tulips and one of the reasons many kinds fail to return year after year. A few varieties like Darwin tulips, which are known as better perennializers, actually do so because they RESIST multiplying; a large first-year bulb may divide into 2 or 3 blooming-sized bulbs, and subsequently the smaller ones will be less likely to split farther.

When to dig them up: when you have lots of leaves and few to no flowers is a good indicator. Mark the spot clearly and come back to the task either in early or full summer.

Sort bulbs into sizes, roughly.

Definitely refresh the soil, using at least two to four gallons of aged compost (you can't tell what was in the compost; it's all brown and crumbly and earthy-smelling) plus a handful of general garden fertiliser to a square yard. Mix it all through the native soil of the patch and replant with the best bulbs you have. You can water the area to settle it, or let the natural rains do so.

This is a task (for me, anyway) in full summer as the bulbs are fully dormant then and they have time to settle before they start putting out roots.

In the ground they will put out roots long before you're seeing bulbs for sale, even in a dry summer. In a cold-winter zone they sort of hibernate until spring once the frosts set in. In warmer zones they keep on growing.

What to do with the surplus bulbs? Discard. Plant out in a nursery patch, in strict rows, to grow on for up to three years to reach flowering size. Label them with a durable marker - name and date planted at the least. Give away to friends for the same purpose (grow on).

If your Narcissus, particularly, have gone to seed before you can deadhead, you might find little sprouts like skinny scallions in the area around the bulb patch. They might be very different from the parent bulbs - and they can take up to six years before they reach flowering size.