The white iPhone 4 is susceptible to the limera1n jailbreak, according to tweets by iH8Sn0w.
Quote (iH8sn0w)
For future (tomorrow or today) White iPhone 4 users, limera1n is NOT patched and has no difficulty with current jailbreaks.
Also for those wondering, i0n1c's untether will work for the device as well.
Quote (i0n1c):
i0n1c's exploit is iOS based. Since the iPhone 4 runs the SAME iOS, its fine.