As far as I know, the tail rotor in helicopters prevents the aircraft from turning in the opposite direction that the main blades turn (due to Newton's third law). On the other hand, torque in single-engine airplanes makes them drift away from the center of the runway during take-off (which must be compensated by using the rudder). Now, my question is: once in the air, why doesn't this torque make the whole aircraft spin along the longitudinal axe? Perhaps the engine's power isn't enough to exceed the resistance of the wings against the air?
First, it's about torque, not power. A helicopter rotor has a large amount of torque at a relatively low RPM. A single engine airplane has much less torque at a higher RPM (Power = Torque X RPM). On top of that, for a craft of similar size, the helicopter will have much more power (it has to lift the copter directly.) So the helicopter has to deal with torque levels several multiples higher than that of a single engine aircraft.
As to the solution, most single engine aircraft have the vertical stab offset by a small amount to generate a counter force to the torque. If you look at some high power aircraft such as the P-51, it's a noticeable offset. And that's set for cruise power. Takeoff power requires additional rudder input.
It should also be noted that for smaller aircraft, P-factor is a bigger turning force during climb than engine torque.
Often times propellor driven aircraft have differential angles of incidence built into their wings such that they generate a counter torque to the engine. Some turboprops also use differential exhaust stacks to make use of reactive thrust from engine exhaust to counter engine torque.
A Cessna 172 doesn't have enough torque to counteract the stabilizing effect of the wings and empennage. The Grumman F8F Bearcat had a reputation for giving wild rides if power was applied too quickly at low airspeed. My understanding is that you could torque roll an F8F ON THE GROUND without moving (much) - just jam the throttle forward from idle and watch how fast 2000 horsepower can pick an airplane up off the ground and flip it over.