The USPS has regulations against mailing canaries. However, some people do mail them. Not only are they illegal to ship, if you are picking up a package with canaries and the post office finds out what you have, they are required by regulations to report the incident and try to prevent further mailings of canaries. See the USPS Regulations. Here's a summary of what it says:

Nonmailable Live Animals

Nonmailable types of animals discovered in the mailstream must be reported in accordance with the instructions in POM 139.118. Additionally, the mailer and the post office of mailing must be notified to prevent future shipments from being incorrectly accepted for mailing.
525.1 Live Birds
Live birds, other than the type of day-old poultry specified in 526.31, are nonmailable. In addition, day-old poultry vaccinated with Newcastle disease (live virus) is nonmailable. The following are some examples of nonmailable birds:
a. Canaries.
b. Cockatiels.
c. Emus.
d. Finches.
e. Parakeets.
f. Parrots.
g. Adult birds, except those permitted under the conditions in DMM C022.3.3 and C022.3.4.

Be aware that the preceding list is not all-inclusive. Generally, all types of birds not listed in DMM C022 are nonmailable.