When choosing compressor you have to consider if it can constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast 
impressive pressure figures but really poor flow rates adequate limited to airbrush pistols. Even if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but stream of air was very weak. It is sufficient if all you want to do is engrave, although even then we got significant smoke traces around engraved Air Compressor For Laser Cutting picture.
it was rated for level of ~2.4cfm and had a 1 gallon tank which may be filled to 58 PSI. It managed to give us constant 1 bar (14 PSI) of pressure, but it had to run constantly and overheat badly within 5 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 can run at constant ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve air flow in tank we are able to also drive up to 3.5bar (50psi) for short jobs.
The best setup in my opinion will be compressor rated to >6cfm, large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.