You can choose to turn the Mailbird new email sound notification on or off. Here's how:
1) Go to the Mailbird menu in the top left of the main window.
2) Select "Settings"
3) Click the "General" tab
4) Click the "New message sound". To turn off, just choose "None".
Custom sound notification:
You can also add your own notification sounds. Please note that the format of the notification sound has to be .mp3 or .wav and maximum size for best result is not more than 2Mb.