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")

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")

5x5: P1 win (TonyK brute force computer search)