At closer distances it is more like a disk extending out from the Kuiper belt, but at larger distances it transitions to a gigantic and diffuse spherical shell, like this:Is it really spherical though or more like a torus? I always envisioned it like a torus because there is the "hole" in the middle (where the planets are.)
Note that the above graphic shows distances on a logarithmic scale -- otherwise you wouldn't be able to see everything at once. The spherical shell part is much bigger and holds far more volume than the disk-like part. The majority of objects in the Oort Cloud are in that spherical region. We know that it must be shaped like that because incoming long period comets have very little preference for inclination. They can come from any direction with nearly equal probability.