If you and your spouse manage all of your finances together, then having joint accounts is usually the best approach. Your right after the death of one spouse depends on the type of joint account.
It is best to have a Joint Tenants With Rights of Survivorship (JTWROS) account. You both own 100% of the account, which continues after a death. However if you have Tenants In Common (TIC) ownership each spouse can leave their share of the account to somebody else, typically as defined in their will. The deceased spouse’s share of the account goes through probate and may be tied up until the estate closes.
If you manage your finances separately it might make sense to have separate accounts. If you live in a common law state each spouse’s assets remain their own if in separately titled accounts. If you live in a community property state (see below) then all assets are owned equally by both spouses except assets originating before the marriage or inherited, and kept in a separate individually titled account.
If you do have separately titled accounts with no other provisions on the account it would go through probate with the inheritor decided by will. If there is no will the intestate laws of your state will decide the inheritor. In most states bank accounts can have a Pay on Death (POD) option added to have such assets transfer automatically outside of probate. You can also designate a beneficiary for a JTWROS account but that is typically through an In Trust For (ITF) provision to indicate it does not take effect unless both joint owners die.
If you do use POD or ITF provisions make sure to keep them up to date and in line with your will and other estate documents.
Community Property States:
• Alaska (default common law, but you can opt in for community property)
• New Mexico