Example stereograms
Here are some stereograms I've made using this tool. They're presented in a little widget which might help you see them.
If you click the stereogram itself, it will fullscreen the widget. If you find it easier to cross your eyes than uncross them then tick the "invert" checkbox. If you're having trouble uncrossing far enough, try reducing the "pattern width" slider. The ideal position for it depends on you and how big your screen is — generally on a larger screen you should reduce the slider. The "3D" slider on the right works the same as on the Nintendo 3DS. Lastly, the "cheat" checkbox reveals the hidden picture so you know what you're looking for.
Utah
The floral texture in this one comes from here, and the depth map scene is from here.
The Time Vortex
The galaxy texture in this one comes from here.
Stonewall
It turns out that pride flags, varying vertically but not horizontally, are really good patterns for stereograms (if you throw some noise on them).
Magic Haj
The depth map in this is rendered by me from a model from here, and the title is Grey's.
Winter
The depth map in this is from here, and the snowy texture is sampled from here.
Sponza
The depth map in this is from here. It is perhaps too detailed to use in this kind of image but it's a really nice depth map and I wanted to include it because I think even the ways it fails are instructive. (To be clear, this depth map was never intended to be used this way so its failures are my fault for dragging it into this situation.)