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 actually poor flow rates adequate limited to 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 (Smallest on the picture), I do not know very well what specifications it has but stream of air was extremely weak. It is sufficient if all you want to accomplish is engrave, although actually then we got considerable smoke traces around engraved Air Compressor For Laser Cutting china picture.
it had been rated for volume 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, nonetheless 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 container which is filled to ~100psi. With this unit we are able to run at continuous ~2bar (31psi) without 
fear of overheating since compressor engine isn’t on at all times. Using reserve surroundings in tank we can also press 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.