Hi there! Welcome to the H/J world!
Pony, National, and International Derbies are the only recognized derbies by the USHJA. Pony derbies are only for ponies and I believe are held at the same height as the division for each section (small, medium, and large). National Derbies are held at 3’ with options at 3’5". International derbies are set at 3’6" to 4’ with heigh options being 4’3"-4’9" in height.
Those are the derbies held at AA shows and AA shows only, or as their own special occasion show.
Some smaller shows have their own versions. Some do a derby at 2’6" or lower, for fun as a class. Some shows do a hunter challenge type class as a money class (I just participated in one in Venice called the Huntfield VET Hunter Challenge). They’ll have differing formats (some with high options, some without. Some with a second callback as a handy course, some with it just being a second round scored like a classic round).
Very few derbies are held on grass, as very few venues have a grass ring anymore. Even then most hunters like to stay in the ring, as it’s a safer option footing wise - especially when weather is concerned. There are some shows that hold their grand prixs out on grass, and a few places have a Jumper Derby type course that has some eventing type elements with jumper jumps as well.
I’d suggest looking up shows in your area and seeing what they offer. If you’re ever looking just to watch stuff, look up Jumper or Hunter derbies on youtube and let the recommended section play its way through!