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 sufficient only for airbrush pistols. Even if small compressor can push enough air it will likely overheat in short while.
Our laser cutter came with little membrane pump (Smallest on the picture), I do not know what specifications it has but stream of air was Air Compressor For Laser Cutting china extremely weak. It is sufficient if all you have to to do is engrave, although actually then we got substantial smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon tank which can be filled to 58 PSI. It were able to give us constant 1 bar (14 PSI) of pressure, but it had to run on a regular basis 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 continuous ~2bar (31psi) without fear of overheating since compressor engine is not on all the time. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for short jobs.
The best setup for me would be compressor rated to >6cfm, huge 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.