None.
Common law exists only if you have signed your name as Mr & Mrs during your relationship, plus had joint checking savings credit cards as Mr & mrs, had wills made out to each other and naming you as a spouse, then you probably could be considered common law married.....anything less then that , you were just 2 Single people shackin up, but
You do have rights if you actually bought things together out of joint checking savings credit cards, then you would own 1/2. Custody and childcare, the parents are always responsile for the caring and paying of their kids, so nothing is changed here.
what is the question you really want to ask?