I tried running shoes when I first got back into playing tennis. Found out that the thick, soft soles provide no support for lateral movement and can lead to ankle sprains (only took 4 weeks to heal). Same for shoes with nubbies on the bottom. They might be great for turf, but not good for hard courts. I found that New Balance cross trainers with a relatively flat sole provided both support and cushion. I also have some K-Swiss court shoes that provide great support, but are a little lacking in the cushion department.
I can't remember seeing any advanced tennis players wearing running shoes (although I have seen some fugly court shoes).
MG Radical OS, VS Mains @62, Syn Crosses @ 58