The question of a character's handeness has come often on alt.tv.simpsons. Here's a loose collection of my own observations on whose left or right handed on The Simpsons:
Homer is right-handed in 7F11,7F12,8F06,8F13,8F15,2F10,2F12,2F13.
Homer is left-handed in 2F05.
Homer used both hands in 7F23.
Marge is right-handed in 7G08,9F03,2F05.
Marge is left-handed in 9F18,2F15,2F33.
Bart is left-handed in 7G02,7G08,7F03,8F03,8F06
Bart is right-handed in 1F20,2F32.
Bart uses both hands in 8F15,8F16,2F22.
Lisa is right-handed in 7G08,7F24,8F01,8F15,2F32
Lisa uses left-handed in 8F24.
Maggie is left-handed in 1F17,2F32.
Ned is *right-handed* in 7F08.
Ms. Hoover is right-handed in 8F15.
Sideshow Bob is right-handed in 9F22.
In 1F17, all of Ms. Hoover's class seemed to be right-handed.
Barney is right-handed in 2F16.
Ralph is right-handed in 2F16.
Some people who might be left-handed (7F23):
Akira Dr. Monroe Mrs. Lovejoy
Apu Dr. Riviera Ms. Krabappel
Barney Jasper Otto
Burns Karl Princess Cashmere
Chief Wiggum Kent Brockman Richie Sakai
Chuck Ellis Krusty Sideshow Mel
Cowboy Bob Mayor Quimby Skinner
Dr. Hibbert Moe The Happy Sumo Master Chef