Running can be cancelled by crouching, which then lets you do any crouching action like down-tilt, smash attack, or dash (Cactuar dash).
Run cancelling is not super relevant for Fox; it is mostly used to approach with down-tilt, which can be very strong in some matchups (namely, vs. Marth) because it has good range and option-selects ASDI down. Very rarely it can be useful to down-smash while running.
Note that Running shine does not involve a run cancel, because you can always use special moves while running; going through crouch is not necessary.
Further note that run cancel up-smashing, while possible, is entirely vestigial; the Jump cancel up-smash is a better technique that achieves the same thing, and you must learn the latter one way or another.