Team Unlock has announced an iPhone 5 untethered jailbreak for iOS 6.1.3 and 6.1.4, but it's unclear whether it's legitimate. You can give it a shot, but it won't be free: Team Unlock is charging $24.99 for the jailbreak.
The iPhone 5 iOS 6.1.3 and 6.1.4 jailbreak is suspect for a number of reasons. For one thing, jailbreak tools that charge money are typically scams and we haven't been able to independently verify whether you can actually jailbreak an iPhone 5 with Team Unlock's tool.
Team Unlock's website doesn't exactly inspire confidence either. The jailbreak can be found here, along with this message from the developers:
"Within just a FEW MINUTES of using our software solution, you will have an Unlocked & Jailbroken iPhone. Whether your iPhone is a 2G, 3G, 3GS, 4,4S or 5 or iPad.
The #1 Unlock/Jailbreak iPhone Software: Used by Apple Employees!
Within just a FEW MINUTES of using our software solution, you will have an Unlocked & Jailbroken iPhone. Whether you have the iPhone 5, 4S, 4, 3GS, 3G, 2G - we can Unlock and Jailbreak it!"
Trusted iPhone jailbreak team Evasi0n has previously indicated that they intend to hold off on any further jailbreaks (leaving iOS 6.1.3 and iOS 6.1.4 users in the lurch) until Apple releases iOS 7. If another team were to release an iPhone 5 untethered jailbreak in the meantime, it would likely mean that the exploits Team Evasi0n is holding on to for the iOS 7 jailbreak will be fixed by Apple by the time it actually comes around.