Stem bulblets usually attach to the underground portion of lily stems at the end of the growing season. They range in size from garden peas to hazelnuts. Oriental and Trumpet lilies make the fewer number of bulblets each year, perhaps only 3 to 10 per stem. Asiatic lilies are more prolific, sometimes covering the stem from the top of the mother bulb to just above ground level. If you notice black "miniature-like" bulblets growing on the stem at the base of each leaf, they are arial Bulbils, and are treated in the same manner as bulblets growing underground.
Bulblets, depending on their size, from Asiatic lilies can flower within two years; Oriental and Trumpet lilies may take a bit longer for more than one blossom per stem.
When you see grass-like growth near the base of a mature stem in spring, try not to break the little stems, or you will harm your future lily "babies." Let them grow in place next to the mother bulb for a year or two. This is the easiest method; letting Mother Nature do the propagating. When it is time to divide the "clump," they are easily harvested.
If during fall transplanting or dividing, you discover tiny bulblets attached to the underground portion of the stem, cut the stem above the original bulb and at ground level to rescue the bulblets. Do not force them from the stem or they may not survive on their own. For valuable cultivars, we like to dig a shallow trench in a protected area of the garden and simply lay the old stem on its side, and then cover everything with two inches of soil. The bulblets will continue to grow, feeding off the rotting stem.