I was executing a power-on stall, and upon stalling the nose dropped and the plane rolled hard right. I promptly applied left rudder and released yoke backpressure to recover, and the plane recovered normally. However, my instructor told me that I should not use rudder when recovering from a power-on stall, even when the plane rolls one way or the the other. This confused me, because a DPE had told me not to use ailerons when recovering from a power-on stall if the plane rolls one way or other other, so I'm not sure how I'm supposed to level the plane if I can't use either rudder or aileron. Also, I had thought the roll was the beginning of a spin, but my instructor told me that it was not and that we were nowhere near a spin. Can someone clarify the use of rudder (or aileron) on recovery from a power-on stall? And if I'm in a situation where my plane is stalled and I'm rolling hard one direction, if that's not a spin, can someone please clarify what a spin is?
Your instructor is correct.
I've never seen a C172 where the POH actually discusses stall recovery besides saying it's 'standard'. I was taught to keep the airplane in balance using rudder in the approach to the stall, and manage any wing drops with rudder in a fully developed power off stall (I'm in the UK, not sure how the US is different). However, that isn't the procedure for a power on stall. The FAA Airplane Handling Handbook section 4-8 on power on stalls states:
The pilot must promptly recognize when the stall has occurred and take action to prevent a prolonged stalled condition. The pilot should recover from the stall by immediately reducing the AOA and applying as much nose-down control input as required to eliminate the stall warning, level the wings with ailerons, coordinate with rudder, and smoothly advance the power as needed. Since the throttle is already at the climb power setting, this step may simply mean confirming the proper power setting.
In a power on stall if you get a big wing drop the rudder is not going to lift the wing, and coarse rudder movements could stress the airplane, see the AA 587 crash. So balance with rudder, but most importantly reduce your angle of attack with as much forward stick as you need.
Interesting discussion on the topic here.
Most notably, the Australian aviation authority updated their guidance to add:
apply rudder to prevent the nose of the aeroplane yawing into the direction of the dropped wing
I believe this recognizes that there is a very short duration in the maneuver where coordinated rudder inputs are not possible yet, but where a spin can be prevented by stopping the yaw using rudder inputs. Almost immediately this is followed by:
ailerons should be held neutral until control is regained, when the wings should be levelled using coordinated inputs
My suggestion right now is do as your instructor says. Next lesson Ask to get a demonstration or try for yourself.
The full answer is to follow the operations manual. Airplane types vary a lot. The 172 is a very benign type as far as they go. A bit of performance has been traded for simple and safe (well) handling. Other types can be quite nasty, given wrong control inputs they bite.
Rudder is not used for recovery but is an extremely valuable tool to control yaw while plane is being unstalled by reducing AOA.
Ailerons should not be used as a roll control reversal can happen if the lowered aileron causes its wing to stall, while the other wing (with the raised aileron) remained unstalled.
Reducing power also makes the situation less complicated by reducing P and engine torque effects. There will be schools of thought depending on the type of aircraft. A very large or heavily loaded aircraft needs power and airspeed. Push the yoke forward first if signs of stall develop. Many times this is all that is needed.
Applying opposite rudder in an incipient spin (ok, its not fully developed yet but let's act now) is not a bad move as the control input to stop the spin is now already in. With a dropped wing the plane will be in a side-slip. Not a major issue provided the plane is promptly unstalled by reducing AOA (with elevator).
Speaking of incipient spin, we also have the incipient stall: the buffet and stick shake. There is also the stall warning horn. Do not ignore these. Be proactive and reduce AOA immediately, rather than falling further and further into trouble.