in “USPS Publication 52, Hazardous, Restricted, and Perishable Mail” and are pretty clear.
In summary it says concealable (safe to assume that means any) air pistols that don't meet the USPS definition of a fire arm can be mailed but require an adult signature from the recipient. Air rifles that do not meet the USPS definition of a fire arm can be mailed, with an adult signature required for any that shoot over 400 FPS. All state and local laws apply. The USPS fire arm definition is the same as the ATF's, so all airguns can be mailed.
Here are the USPS links to the pages that apply – The regs have had minor changes over the years and I re-check them every 6 months or so. Sec. 431.6, 432.1 a, 432.2 c & e, 432.3 g, and 433 are the ones that apply to airguns. 433 applies specifically to postal workers who tell you you can't mail an airgun.