Running in the wet is never fun. Can't say I've had an issue where my shoe starts holding lots of water and gets spongy. We don't get a whole lot of snow down where I'm at, but we're starting to get into our rainy season.
Time for rain-gear! Tights and jacket here we go!
I already took my winter-gear out of the wardrobe but I've never had running shoes for rain (like the Nike SP) and never felt the need for it. I will try to run with other shoes in the rain to see if there is a big difference.