The following actions can be done while in "Avatar Customize" screen.

Click and hold Right Analog-stick, then move Right Analog-stick: Make Avatar burp continuously in various pitches.
Click Right Analog-stick: Make Avatar burp.
Hold LT or RT: Spin Avatar making it lose its balance.
Hold Right Analog-stick Left then move Right: Make Avatar enthusiastically spin to the right.
Hold Right Thumbstick Right then move Left: Make Avatar enthusiastically spin to the left.
Move Right Analog-stick: Rotate Avatar's head.
Move Right Analog-stick Left, Right, Left, Right: Make Avatar angry.
Move Right Analog-stick Up, Down, Up, Down: Make Avatar happy.
Press LB: Have Avatar do a random expression: depressed, apologetic, surprised, laughing, or angry.
Press RB: Open Avatar's mouth.
Rotate Right Analog-stick in a circle: Make Avatar dizzy.

The following action can be done in the "My Features" area or anywhere the Avatar head is zoomed in on.

Move Right Analog-stick Up then Down: Make Avatar hit its head on the screen, making it dizzy.

The following actions can be done while in Xbox 360 Dashboard.

Click Right Analog-stick: Make Avatar smile and wave.
Move Right Analog-stick: Rotate Avatar's head.
Move Right Analog-stick Left, Right, Left, Right: Make Avatar angry.
Move Right Analog-stick Up, Down, Up, Down: Make Avatar happy.
Rotate Right Analog-stick in a circle: Make Avatar dizzy.