Precious/semi-precious stones are rather arbitrarily defined. Various forms of chalcedony (agates, jasper, chert, etc) can often be called such, and are very common sometimes.
I’ve buckets of agates and jasper, among other things (like petrified wood) that I’ve found in fields while doing my ag job here in the willamette valley of Oregon. Oregon as a whole is amazing for rockhounding. There are some spots where I can’t go a several feet without seeing a 1cm or larger agate just laying there. Makes it hard to concentrate. I’ve mostly got it out of my system now though.
Cleaned up a bunch thanks to an ultrasonic cleaner, and one day will tumble a bunch to give away.
Yeah, I dug up a bunch of quartz crystals, some granite containng small fossils (tiny clams, small spiral shells etc), and a few pieces of turqoise in the backyard as a kid. How interesting the rocks in your own backyard are depends a lot on the geological history of the area though.
