I had some luck in the past with various planes being positioned in the scene, with a scrolling rain texture applied to them. You can fade the texture out at the bottom (a la soft particles) so that you don’t see where the planes enter the ground, and you can fade them out as you approach them or look at them from a grazing angle so that you don’t ever get too close to one of them.
That would work for rain; I don’t think it would be a good option for cats and dogs.
Then you could consider a postprocessing shader - I haven’t actually tried this, but in theory it should be possible to do this in post since the depth buffer alone should give most of the information you need. You would also need some sort of occlusion map or shadow map or other way to determine where the rain falls and doesn’t fall, though.