It will take more patience and tending on your part, but for shady areas you'll get better results with seed. Sod is great when you want instant lawn. You get grass that's thick, weed-free and fertilized. You can be mowing the stuff in two to three weeks.
But most sods are grown in wide-open fields, a condition that favors bluegrass and other sun-loving grasses. And sod that's primarily bluegrass won't grow as vigorously under your shade tree as fescue and other shade-loving grasses. Head to a landscape center and buy a grass seed mix that's formulated for shade. Some mixes get pretty specific in their formulation, so if you know what kind of soil you're working with, it will help you find the ideal mix. And you'll save money—grass seed for a 100-sq.-ft. area will only cost a few bucks, while sod could cost ten times as much, or more. Plus, it's a lot easier carrying home a sandwich bag of seed than hauling a dozen rolls of sod.
Your seed will require more soil preparation and pampering. You'll need to rototill or loosen up the soil before planting, keep the area watered and battle weeds as they try to take root before the grass fills in. And you won't be playing croquet for at least six months. But once it's established, seed has the best chance of survival.