Rules of image's separability
A separable image $I$ that could be written in a separable way with vectors $a$ and $b$ as:

$$ I= a^Tb$$

would necessarily be of rank one at most. Either a zero-matrix (rank 0), or with only one linearly independent row or column. This are rather dull images alone. But this could be a model for a slowly varying background, some people use that for background source separation in image sequences. The following images are such images, generated the following way. $a$ and $b$ are uniform random sequences, $a_s$ is $a$ with sorted values, and  $b_s$ is the cumulative sum of vector $b$:

 

However, splitting an image in a weighted sum of rank-one matrices is at play in SVD decomposition. Finding low-rank approximations to matrices is a very current problem, and depends a lot on "how close is the approximation" (ie: with what norm?). 


Wikipedia: Low-rank approximation
Bart Wronski: Separate your filters! Separability, SVD and low-rank approximation of 2D image processing filters