For what it's worth, here are a summary of the answers so far for the Misere game:
1x1: P2 win (poisoned chalice)
2x2: P1 win (P1 makes an arbitrary move)
3x3: P1 win (P1 plays in the centre, and his 2nd move is a knight's move away from P2's first move, and then solve by hand: see Kristal Cantwell's answer).
4x4 and more generally 4n x 4n: P2 win (strategy stealing: see gowers' "better approach")