I have a fairly good set of plastic tweezers in my first aid kit that I use for cactus spines; that and duct tape for the little fuzzy spines called glochids that come with prickly pear. I carry a comb and good forceps when I am out with my dog.

Around here, the ouch-champ is the so-called jumping cholla. Spines from that can easily go in a half inch and they are barbed making them a real bi+ch to pull. I once had one go through a tendon on my right hand and into the bone underneath. I couldn't move one finger until I pulled the spine that had things nailed together. Bad stuff!