When choosing compressor you need to consider if it can constantly provide desired pressure and level of air. Little airbrush compressors small compressors boast amazing pressure figures but actually poor flow rates enough only for airbrush pistols. Even if small compressor can push enough air it will likely overheat in short while.
Our laser cutter was included with small membrane pump (Smallest upon the picture), I really do not know what specifications it has but blast of air was extremely weak. It is sufficient if all you want to accomplish is engrave, although even then we got significant smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon container which may be Air Compressor For Laser Cutting filled to 58 PSI. It managed to give us continuous 1 bar (14 PSI) of pressure, but it had to operate 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 are able to run at constant ~2bar (31psi) without fear of overheating since compressor engine isn’t on all the time. Using reserve atmosphere in tank we can also drive up to 3.5bar (50psi) for brief 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.