This is what I've been doing for years:
When you pay a bill, keep that statement with the check #, amount, and date paid written on it. When you get the next statement, make sure that everything is correct and shred the previous month's bill.
This will work with all statments: medical bills, credit cards, utilities, etc.
If you pay off something -- a revolving credit card, a loan, a medical bill, etc. -- keep the letter or statement that the company will send you that says that the account is paid in full. You may need this letter or statement in the future in case the company ever comes to you saying you owe them money. Also, your credit rating can sometimes be affected by these things, but if you have proof that it has been paid in full, it makes things a lot easier when it comes to correcting your credit report if needed.
If your talking about your Explanation of Benefits that medical and dental insurance companies send you, I do these a little bit differently. I will usually get a statement from the medical/dental office where the visit took place. If I get an EOB from the insurance company, then I get a statement from the office for the same visit and the statement says that all has been paid up and we're good, then I throw the EOB away and keep the statement from the medical office. kwim? If I don't receive a statement from the medical/dental office, then I hold onto the EOB in case a dispute ever comes up. I don't know if this is the 'necessary' way to do it, but this is what I do.