Good Korean radishes are firm and the skin is a little shiny, without any scratches. The top should be a little green. The photo below is a is a good example of quality Korean radishes you should be choosing.
Korean radish is a varietal of the Daikon radish but it’s very different than the Daikon you might find in the store. It’s shorter, stouter, and has a pale green shade halfway down from the top. The flesh is denser than a Daikon and the leaves are a lot smoother. Be careful not to buy a Daikon by mistake!