Our bank issues both ATM cards and debit cards (ATM & POS). Are there regulations concerning the type(s) of accounts that either of them can be tied to?

Check your network rules for one potential source of limitations. In general, a debit card has to be tied to a transaction account as its principal account for POS transactions, although it can be permitted to access savings and MMDA accounts at ATMs. Even with those rules, however, it's often possible (although frowned up) to make an MMDA the primary POS account. The problem, of course, is the restriction on savings account transactions in Regulation D.