I used a clean 9V power source (not a battery) and arranged a simple circuit on my breadboard with a potentiometer in series with a 1k (1%) metal film resistor and the diode. Varied the pot to provide a range of currents. Measured the voltage drop across the resistor to get the current and the voltage drop across the diode. Recorded everything by hand - the old-fashioned way!
As I mentioned, I'm going to ultimately get some MA856s and BAS33s to check out, and do some extensive listening in my test circuit. I really don't think there will be a tremendous difference in tone, but it will be fun trying out the different diodes.
By the way, my interest in all of this comes from a recent purchase (on a whim) of the Tone City King of Blues pedal. People were touting it as a KOT clone, but in fact it turned out to be a Klon (Engine A) into a standard Blues Breaker (Engine B) circuit. Everything (almost) was SMD, including the BB silicon diodes! (The Klon diodes were full size 1N34As). In the end, the BB side of the pedal sounded very blah to me, so I thought I'd have look at the KOT and BB circuits (and the many derivative pedals out there) to see if I could improve the sound of the BB, with aim of making my own DIY King of Blues dual pedal.