As a children’s book illustrator I have attempted to depict dragons many times.
Because my style veers towards the literal, this causes me no end of stress as I can’t seem to get my pencil round the anatomy of a creature that enjoys the use of four legs as well as a pair of massive wings. I spend a lot of time arranging the composition so that this anatomical absurdity is cunningly disguised, usually with liberal swathes of billowing smoke and mist. Occasionally, I discover the text does not stipulate the number of appendages and I am free to opt for my preferred dragon physique: that of a bat or pteroadactyl. The downside to this approach is that the poor creature is reduced to a shuffling mess when earth-bound but, for elegance in flying, the loss of two legs is worth it; I can avoid getting in a muddle with my muscles (the pectorals and deltoids being particularly awkward).
Recently I have been drawing a lot of dragons for a series of book covers (sketches and one finished example attached, as well as a couple of previous takes on the beasties in question). The other day I was moaning to Terri Windling in the butchers about this subject, and she has instigated a moveable feast over on her blog concerning the nature of dragons in mythology.