How do I prevent the context menu from showing up on certain keyboard keys (like Keys.Apps)?

Override WndProc in your Control and do the following. You should then listen to keyup and show the context menu yourself.

[C#]
protected override void WndProc(ref Message m)
{
	if(m.Msg == 0x7b /*WM_CONTEXTMENU*/ )
	{
		return;
	}
	if(m.Msg == 0x101 /*WM_KEYUP*/)
	{
		Keys keys = (Keys)m.WParam.ToInt32();
		// Prevent this key from being processed.
		if(keys == Keys.Apps)
			return;
	}
	base.WndProc(ref m);
}

[VB.Net]
Protected Overrides  Sub WndProc(ByRef m As Message)
	If m.Msg = 0x7b Then ’WM_CONTEXTMENU
		Return
	End If
	If m.Msg = 0x101 Then ’WM_KEYUP
		Dim keys As Keys = CType(m.WParam.ToInt32(), Keys)
		’ Prevent this key from being processed.
		If keys = Keys.Apps Then
			Return
		End If
	End If
	MyBase.WndProc( m)
End Sub