Yes you are allowed to spin at least some model C172's when flown in the utility category, but they are a dreadful aeroplane to teach spinning in. You need pro spin aileron and they come out of the spin the minute you relax any control inputs.
The only 172s which are not permitted spinning, when properly loaded, are the172RG, and float equipped 172's. All others include spinning within their limitations.
172's are not aerobatic aircraft, and were never intended for aerobatic training, thus their tendencies in a spin are optimized for safety, not aggressive or demanding handling (you can't have it both ways). They will spin and recover nicely, with conventional recovery techniques, and are thus adequate for training spins, and certainly far superior for teaching spins to a spins prohibited 'plane! Yes, you can mishandle a 172 into a spin with ailerons, though they will enter and recover nicely with no use of ailerons at all. An unmodified 172 will not recover from a spin until recovery is initiated, though once recovery is initiated, it is rare for aggressive recovery control input to be needed to affect recovery.