They're not as sexy as morels, nor as predictable, but oyster mushrooms -- Pleurotus ostreatus -- might just be my favorite wild mushroom. And I don't even like oysters.
The stalks of oyster mushrooms are very short, and they don't come out of the middle of the caps. Instead, the stem is off to the side, allowing the mushrooms to grow in overlapping clusters. In fact, their genus name, Pleurotus, comes from the Greek word for side. (And I always thought it was related to the French word for rain!)
Triplax lay their eggs on oyster mushrooms, and spend the rest of their lives hanging out among the gills and slowly eating them. They're harmless and easy enough to wash off, and a great identification aid.
Which is important, because there are other similar white mushrooms that grow on wood. They probably won't kill you, but they're supposed to taste awful. Luckily, oyster mushrooms are frequently available at farmers' markets and at high-end grocery stores. You can stop there, or you can use cultivated mushrooms to get a good sense of what they're like there before you go out looking for them (though the ones you see in the store will be relatively dried up after several days of storage, and some farmers select for more colorful strains). Your absolute best bet for safely identifying wild mushrooms is to out with experienced mushroomers, like the ones at the Mycological Association of Washington. (We sometimes find mushrooms on our wild food walks as well.)
In your yard (or apartment): Oyster mushrooms can be cultivated on many different substrates that contain cellulose -- not just logs but also shredded paper, coffee grounds, or sawdust. There are even kits using rolls of toilet paper as the base, designed to grow indoors. If you want to give it a try, check out the products at Field & Forest or Fungi Perfecti.