"Je m'excuse" should be avoided. It's rather used in a disagreement.
Excusez-moi is used for treading on someone's foot : ) or getting someone's attention.
Desole and pardon can also be used to appologize.
Pardon can be used to shove your way through a crowd. : )
Desole can be used when you can't help someone. (If you don't have the time or a lighter for instance)