Anecdotally the bare minimum is hips/elbows.
The real serious breeders are following the gsp club recs of hips, elbows, eyes, heart, cone degeneration. Extras are von willebrands disease (blood disorder), and autoimmune thyroiditis.
Luckily shorthairs are still a pretty healthy breed so the risk is lower with skipping some of these tests.
I think we're starting to see more shorthairs that aren't as naturally gifted hunters so I'd be more concerned about the pedigree, as the hunting/field trialing breeders are usually using pretty healthy genetics at the moment. They also can breed pretty high powered dogs beyond what most of us can handle...
black shorthairs are becoming a little more prevalent and bringing in more diversity, but maybe eventually a problem since any dog with sharp shooters man in black (navdha/versatile club lines) in the pedigree seems kind of popular as a stud.
