I've been told not to use synthetic because it reacts with plastic globes on filters and weakens them. A friend of mine worked in a big shop that got a new compressor, it said in the instructions not to use synthetic oil, but the tech from the place that sold it to them put synthetic in it anyway, I don't know if this was after a breakin period or not. Two days later another tech was pulling the compressor and installing a new one, the other tech was no longer with the company. I forget the size, but it was a big screw type or rotary compressor, many thousands of dollars.
I called DeVilbiss, the manufacturer of my 2 stage Delta compressor I got at Sam's Club. The instructions talked about 30W compressor oil, which I couldn't specifically find locally, although I didn't check any commercial compressor shops. They said 30W non detergant oil was fine.