Here's an experiment in distorted perspective for a background for a work in progress, my version of The Gingerbread Man story. This was done by eye, with no rulers or curves or computers, aside from the Bakery sign, which was done using free-transform in photoshop. It's a technique inspired by Salgood Sam and Kim Jung Gi, two artists who excel at this kind of thing.
It's a technique often used in animation. Salgood as a great theory about this use of perspective involving how we see binocularly, and that one-point perspective is a composite illusion that doesn't necessarily reflect what we're actually seeing. Of course, this is much more exaggerated.
There are certain consistencies--the gridded floor follows a kind of contour, but there are multiple vanishing points going in multiple directions, with the landscape curving wherever I want it to accommodate the space. I wanted to squeeze as much as possible in the image as I could. This distortion also reflects the tone of the piece, a wonky doughy whimsical world with a lot of action and vitality.
In the past I've been very self-conscious about "getting it right." Now that I've figured out essentially how to do it right, I feel more comfortable veering off that path. Similar techniques are sometimes used in animation. As long as it all has its own internal logic and your distortions are purposeful and not accidental, there's a lot you can get away with.