When choosing compressor you have to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors small compressors boast impressive pressure figures but really poor flow rates adequate only for airbrush pistols. Even if small compressor would be able to push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Air Compressor For Laser Cutting Smallest upon the picture), I do not know what specifications it has but stream of air was extremely weak. It is sufficient if all you have to to accomplish is engrave, although actually then we got substantial smoke traces around engraved picture.
it had been rated for level of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to run all the time and overheat badly within five minutes.
Third and current compressor we got was 6cfm rated belt driven device with 13 gallon tank which is filled to ~100psi. With this unit we are able to run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on constantly. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for short jobs.
The best setup for me will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.