The only way to grow lawn grass from clippings is to allow the grass to grow so long it produces seed. Then, when the seed remains in the clippings and conditions are right, it may germinate. Allowing grass to become that overgrown looks unkempt and may violate regulations in your community. Other grass propagation methods produce more reliable results.
Turf grass can be grown from seed, by planting plugs of rooted grass or allowing some grass types to spread by stolons or runners -- stems and roots that form new plants. Another technique, sprigging, involves planting stolons with nodes, much the same way as plugs. The success rate of each method depends on the type of grass and growing conditions. Plugs do not detach and become components of clippings, though stolons might. The chance of stolons surviving mower blades and rooting successfully is minimal.
Some grasses establish better or more quickly from seed than others. Annual ryegrass (Lollium multiflorum), often included in lawn seed mixes, germinates and sprouts quickly. If mowed after going to seed, the seeds might germinate, given enough sun and water. Annual ryegrass is usually included in mixes, rather than sold as a stand-alone product, because it produces clumps, rather than a fine, smooth lawn. It won't live for more than a year, but allowing the grass to set seed would produce at least some new grass plants every year.
Kentucky bluegrass (Poa pratensis), which grows in areas that roughly cover U.S. Department of Agriculture plant hardiness zones 3 through 7, is a perennial grass that might grow from seeds contained in clippings. It is also a longtime favorite for lawns. Bluegrass is widely sold, often in mixtures of several varieties, and produces a fine, smooth lawn. Though slow to establish, bluegrass has been reported as invasive, suggesting seed germination rates are high.
Grass Clipping Considerations
If you're looking after your lawn, you mow it when it gets to no more than 3 inches tall, which is before it starts to set seed. Standard grass clippings without seed help return nutrients back to the soil if you use a mulching lawn mower and leave them on the lawn. The clippings degrade quickly, adding organic material to the soil. Collecting and composting grass clippings, along with other organic wastes, will produce finished compost that is useful for amending and enriching soil before grass seed or plugs are planted.