When choosing compressor you have to consider if it can constantly provide desired pressure and volume of air. Little airbrush compressors little compressors boast impressive pressure figures but really poor flow rates sufficient only for airbrush pistols. Actually if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with little membrane pump (Smallest upon the picture), I really do not know very well what specifications it has but blast of air was extremely weak. It is enough if all you want to do is engrave, although even then we got substantial smoke traces around engraved picture.
it was rated for volume of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It managed to give us Air Compressor For Laser Cutting constant 1 bar (14 PSI) of pressure, but it had to operate constantly and overheat badly within 5 minutes.
Third and current compressor we got was 6cfm rated belt driven unit with 13 gallon tank which is filled to ~100psi. With this unit we can run at continuous ~2bar (31psi) without concern with overheating since compressor engine is not on all the time. Using reserve surroundings in tank we are able to also force up to 3.5bar (50psi) for short jobs.
The best setup for me will be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.