Go backwards:  Press Z while jumping.

Moon Gate level:  Successfully complete the Do Not Enter challenge.

Helidrop level:  Successfully complete the Ski Patrol Chief challenge.

Costume 2:  Get at least 50,000 points.
Costume 3:  Get at least 100,000 points.
Costume 4:  Get at least 150,000 points.
Costume 5:  Get at least 200,000 points.
Costume 6:  Get at least 250,000 points.
Costume 7:  Get at least 375,000 points.
Costume 8:  Get at least 500,000 points.
Costume 9:  Get at least 625,000 points.
Costume 10:  Get at least 750,000 points.
Costume 11:  Get at least 875,000 points.
Costume 12:  Get at least 1,000,000 points.
Costume 13:  Get at least 1,250,000 points.
Costume 14:  Get at least 1,500,000 points.
Costume 15:  Get at least 1,750,000 points.
Costume 16:  Get at least 2,000,000 points.

Dynamic 3000 snowboard:  Get at least 1,000,000 points.

Prototype MG snowboard:  Get at least 2,000,000 points.

Slick 250 snowboard:  Get at least 250,000 points.

Boost of air:  When on an area that has a bomb, jump immediately before it then land on it. This requires some practice. When timed correctly, you will be able to get a quick special move (such as A + Y + A) in for an extra 6,000 or so points. This is useful for beginners that do not have all of their equipment yet.