When choosing compressor you should consider if it will be able to constantly provide desired pressure and Air Compressor For Laser Cutting volume of air. Small airbrush compressors little compressors boast amazing pressure figures but really poor flow rates adequate only for airbrush pistols. Also if small compressor would be able to push enough air it’ll likely overheat in short while.
Our laser cutter came with small membrane pump (Smallest on 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 actually then we got substantial smoke traces around engraved picture.
it had been rated for volume of ~2.4cfm and had a 1 gallon container which may be filled to 58 PSI. It were able to give us continuous 1 bar (14 PSI) of pressure, but it had to operate 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 concern with overheating since compressor engine isn’t on all the time. Using reserve atmosphere in tank we are able to also drive up to 3.5bar (50psi) for short jobs.
The best setup for me would be compressor rated to >6cfm, 
large 70+ gallons tank rated to 10bar (140psi) and software air control via solenoid valve.