To be clear, a 'free' upgrade is very unlikely to happen. You need to be prepared to fork over either miles or cash to do so, and your ability to do either will depend on a combination of the fare you've bought and your status in either MPC or M&M. It is not useful to give highly general advice in the absence of a specific example.
One point that is generally true for most airlines is that the cheap economy tickets are not upgradeable using miles, and if they are will only be upgradeable to the next class of service (in the case of LH and CX, premium economy).