Looking at the C172S POH it does actually say in the normal procedures section for short field takeoff with flap 10:
6. Elevator control - SLIGHTLY TAIL LOW
7. Climb Airspeed - 56 KIAS (until all obstacles cleared)
8. Wing Flaps - RETRACT SLOWLY (when airspeed is more than 60 KIAS)
For a normal takeoff:
4. Elevator Control - LIFT NOSEWHEEL AT 55 KIAS
5. Climb Airspeed - 70 - 80 KIAS
6. Wing Flaps - RETRACT (at safe altitude)
The expanded soft field procedure:
Soft or rough field takeoffs are performed with 10° flaps by lifting the airplane off the ground as soon as practical in a slightly tail low attitude. If no obstacles are ahead, the airplane should be leveled off immediately to accelerate to a higher climb speed. When departing a soft field with an aft C.G. loading, the elevator trim control should be adjusted towards the nose down direction to give comfortable control wheel forces during the initial climb.
There's actually a surprising amount of other useful information in that POH. Cessna have definitely put a lot of work into making it useful. The C172N POH is not as clear and it doesn't recommend using flaps at all, and it has slightly different target airspeeds.