My experiences:
I was taught spins in an aerobat with 'chutes, and that's the way I would prefer to teach them if asked. (Something aerobatic and with chutes.) At the college I used to work at, we had a dozen or so C-150/152s and a single PA38. One older 150 was the designated "spin bird", because they didn't want to keep buying gyros for the rest of the fleet. Instructors/students caught spinning one of the newer birds could expect to cough up the cost of an overhauled AH if they returned to the ramp with one tumbled.
We were prohibited from spinning the PA38 as well. I most certainly remember the tail wagging quite eagerly during stalls. IIRC, there was an AD for an extra brace on the vertical stab spar because of cracking on the first year models. Later years had the brace installed at the factory. It was an excellent X-country trainer though, more comfortable than the Cessnas for larger folks. I believe the stall/spin accident rate for the PA38 is double that for other common US trainers according to NTSB.
Though it never happened to me in my training, when I started instructing, I did learn that just when you thought a 150 was as tame as an elderly housecat, it would occasionally flick inverted during accelerated stalls.
One drill my original instructor used, which has stuck with me to this day is to have the student put hands in lap while instructor apply (nearly) full aft elevator and neutral aileron. Have student keep wings approximately level with rudder alone. Aircraft gently bucks and rocks in/out of a mild power off stall. Most benign GA aircraft won't spin if you don't let them yaw. We did this in a J-3 while slowly descending 2-3000'. Works well in the Cessnas, sometimes requiring almost full rudder to maintain wings level in turbulence.