I'm wondering if there is a rule that covers the use of capital letters for Mum/mum

If you are referring generally, say, to an unmarried girl who has a baby, then she is 'a single mum'.
If you are referring to someone else's mother, then, "Will your mum let you go?"

If you are referring to your own mother, then treat is as you would a real name and capitalize it: Mum.