Spent the better part of Saturday pulling my fridge out and following this blue residue that kept showing up on the baseboard near the compressor. I was convinced the compressor itself was leaking something, wich had me pretty worried. Wiped it down twice over the last month and it kept coming back, always in the same spot.
Turns out the culprit is the little water line running up to the ice maker. Where it connects down near the bottom, there's some leftover soldering flux from when the line was put in years ago, and moisture wicking along the tube carries it out and down the wall. The blue color matches it exactly once it dries. It's not from the fridge at all.
Cleaned the whole area with vinegar and water, dried everything, and tucked the tube so it drips away from the wall instead of onto it. I'll check again in a week or two, but so far the baseboard is staying clean. If you've got blue staining near your compressor, look at the water line first before assuming the worst like I did.