Before and during startup: set and backed up by manually holding the brakes
During taxi: off if you need to stop to give way to another aircraft, ON (this is a must) on all holding points, since it's quite hard (especially for a student) to recognize that the aircraft is slowly moving - and once you pass the holding line, it's officially runway incursion. Setting the parking brake at arriving on holding point and releasing it when cleared for lineup shouldn't be a matter of checklist, but it has to be done automatically. Far from saying the parking brake item shouldn't be on the checklist - on the opposite, it should be included as the last item, but parking brake should be released automatically. If not, it's only a matter of time when the future pilot will be "in a hurry", skip over the whole or at least part (we prefer to skip the last parts) of it, forget to release the brake and the party begins. However, parking brake should not be set once past the holding point (e.g. on the runway during line-up), since you never know when you'll have to immediately vacate the runway (for example aircraft with engine failure on opposite final) - besides, the time between line-up and takeoff clearance isn't longer than 20-30 seconds. It's not that hard to step on the brakes for a minute or so when you don't have anything else to do, except to scan for traffic and listen to ATC awaiting takeoff clearance.
Shutdown: on, then released after the engine stops, since the brakes tend to get quite warm during landing (especially if you had to brake quite hard) and taxi (if you had to use differential braking). I wouldn't set the parking brake on unless the apron has a significant slope - and even then it's FAR better to use chocks.