As per the title. But I would also like to know if both (Engine and Wing) are turned on at the same time or if different conditions dictate the use of each one. I am specifically interested in the case for the B737NG.
In general the cowls get operated way more than the wings, because the operating requirements are more stringent, because ice can form on cowl inlets more readily than the wings in conditions near freezing (temperature drop from inlet flow) and the consequences can be worse when the ice is shed.
Can't speak for the 737, but on the CRJs at least, the cowls have to be on in conditions where the wings don't, such as flying in visible moisture above 230 kts (above which the wings only have to be on if you get an actual detection of ice on the airframe). So when you depart in real icing or "technical icing" (visible moisture below a certain temperature) conditions, both wings and cowls will be on, but during the enroute climb you don't need the wings on without an ICE caution message, whereas you still have to put the cowls on pretty much any time you are in cloud.
In the CRJ program the duty cycle of the cowl AI system was several times higher than the wing AI system because of this, and I'm sure most airliners are similar.
The Boeing 737 FCOM agrees quite well with the procedures John K showed for the CRJ:
- Engine anti-ice should be on when flying in icing conditions.
- Wing anti-ice should only be turned on when actual icing has been detected.
Engine Anti-Ice Operation - In-flight
Engine anti–ice must be ON during all flight operations when icing conditions exist or are anticipated, except during climb and cruise when the temperature is below -40°C SAT. Engine anti–ice must be ON before, and during descent in all icing conditions, including temperatures below -40°C SAT. [...]
Wing Anti-ice Operation - In-Flight
Ice accumulation on the flight deck window frames, windshield center post, or on the windshield wiper arm may be used as an indication of structural icing conditions and the need to turn on wing anti-ice.
The wing anti-ice system may be used as a de-icer or anti-icer in flight only. The primary method is to use it as a de-icer by allowing ice to accumulate before turning wing anti-ice on. [...]
(Boeing 737 NG FCOMv1 SP.16.9 - Supplementary Procedures - Adverse Weather)