A dog with flaky skin will have lots of fine particles of dandruff (basically, clumps of dead skin cells) appearing in their fur. Take a look in their bed or areas where they spend lots of time, as these are likely to contain evidence of their shed skin.
Dry, flaky skin can be incredibly itchy so you might notice your dog scratching or licking him or herself more than usual (see Itchy Skin). Rolling around on the floor or rubbing themselves against furniture is another sign of discomfort in an itchy dog. Too much scratching can also bring on hair loss so you may start to see the odd bald patch cropping up. Scabs and inflamed areas may show up where the skin has been broken and possibly infected by the scratching. You’ll know if the skin is infected as it’ll give off a bad smell.
A dog with unhealthy skin is also likely to have poorer coat condition so you may find the coat is dull and generally lacking its usual lustre. As itching can be distressing, an excessively itchy dog is likely to be an unhappy one so look out for any changes in their mood.