Tom,
I could say "quite accurate", but it, mostly, depends on the shooter.
I have reloaded for it, my handloads are superior to most factory ammo, but the 15 digests all 38 special loads well.
You will discern, through experimentation which load works best in your gun, and for your shooting skills and I would suggest beginning with a 158 grain bullet. As far as I know thats the standard for regulation for an S&W 38 special revolver.
For plinking, the American Eagle 130 gr ball ammo is as good as any at regulated range distances.
The model 15 was named "Combat Masterpiece" before model numbers came along, I would venture that any 15 or "pre-15" in good condition is more accurate than most shooters and will not inhibit your accuracy in any way.
Mike