Star / Fortune cookies, 3 times a day:
(1) 1 minute after morning wakeup
(2) 1:00 PM
(3) 4:00 PM
The number of regular letter mail you receive per day is random. Some days you may receive 5 or 6, other days only 1. Delivery times are 30 minutes after the hour, from 9:30 AM to 5:30 PM (perhaps as late as 6:30 PM). (Meaning, 9:30 AM, 10:30 AM, 11:30 AM, etc.) You may not get mail at those times, but if you get mail it will be at one of those times.
Special delivery [!] - times depend on your life cycle of your tama character, such as the time you hatched it, and how long it was paused.
Shop restock times: 12:00 AM (midnight), 3:00 PM, 7:00 PM
Matchmaker (when your character is old enough): 10:30 AM, 3:00 PM, and 7:00 PM