Three places, in order of trust

Where	Good for	Watch out for
1. The receiving bank's own instructions (its website, app, a wire sheet or your statement)	The exact code and the extra details the bank wants, such as an intermediary	Use the page for incoming international wires, not a general page
2. Swift's BIC search	Confirming that a code exists and which institution and address it belongs to	It does not tell you which of a bank's codes to use for your payment
3. The person who will be paid	Their own bank's code, copied from their bank	Details sent by email can be changed by a fraudster: confirm by phone

The format checker on the home page comes before all three. It catches typos and impossible codes in a second, but it cannot say who a code belongs to.

What Swift's free BIC search is

Swift's terms say BIC Search gives access to data published in the BIC Directory, the list of codes registered under ISO 9362. Each entry has the institution's name and address (Swift). The same terms say:

It is aimed at occasional users and is currently free, without registration. Swift reserves the right to add registration or payment later.
Results reflect the last BIC Directory publication, which is updated quarterly. A brand-new code may not show yet.
It is provided "as is", without a warranty of accuracy.
Systematic queries to copy all or large parts of the directory are not allowed. That is why sites like this one cannot simply republish it.

Why a bank can have several codes

One institution can hold more than one BIC (Swift FAQ). Bank of America is a live example: BOFAUS3N for wires in US dollars and BOFAUS6S for foreign currency, each with its own address (Bank of America). A directory search on one bank name can therefore return more than one code, and only the bank's instructions tell you which one to use. The same goes for 11-character codes: U.S. Bank gives USBKUS44IMT, branch IMT, for international wires (U.S. Bank).

And a bank can use someone else's code

USAA's instructions for incoming international wires give IRVTUS3NXXX, the code of The Bank of New York, as intermediary (USAA, PDF). A directory search for "USAA" would not lead you there. Only the bank's own sheet would. See USAA's SWIFT code.

A code from an email or invoice

A real code proves little. Swift's FAQ says a BIC gives no guarantee about the organisation's activities (Swift FAQ). A payee's bank code can be genuine while the account number or name next to it is not, so check the whole set of details:

Run the code through the format checker.
Look it up in Swift's BIC search and compare the institution name with the bank the payee says they use.
Call the payee on a number you already had, not one from the email, and read the details back.
For a large or first payment, use the red-flag check.

Big US banks, already looked up

For Chase, Bank of America, Wells Fargo, PNC, USAA, BMO, U.S. Bank, Capital One and Navy Federal, the codes from each bank's own pages are on US bank SWIFT codes.