I have tried most solutions for compressed air, and in the end I got a purpose built airbrush compressor, that I I've modified to fit my needs.
First, I borrowed a friends home built silent, refrigerator type compressor. It worked absolutely perfectly. If you are above average handy with tools, this is a very good way to go. Second hand refrigerator compressors are very cheap, and coupled with the parts from a garage type compressor, it is an easy and inexpensive way to get a silent setup.
Then I had a couple of years with a Testors airbrush compressor. It was probably the worst experience. Almost as loud as a garage compressor, no tank, no reliable regulator and not exactly cheap either.
After that, I went CO2-tank for about a decade. CO2 works perfect for airbrushing, and is the most silent alternative. But the drawback was that it tended to make me be conservative with the use. I always tried to find reasons not to use it, to save on the gas. In the end it made me not to evolve in my airbrushing. To be good at what you do, you need to experiment and practice. When you know that every time you open up that valve, it will cost you, you stop doing it. I'm thinking of getting a CO2 setup again, but mostly as a portable air source.
Then I had a couple of different piston type airbrush compressors, and a garage type compressor. But living in an apartment made the noice a reason for not firing up the compressor, and I was back at not getting the needed experience.
Now I have a Sil-Air silent compressor. I love it, and I think it was worth every cent. I have made some changes to fit my needs, but I could have lived with it as it came, if I wasn't especially picky about my stuff.