Evad3rs are one of the best known of jailbreaking teams and they offered the jailbreak for iOS 7. They should be able to jailbreak iOS 7.1 but at the moment we have only heard from Winocm.
The hacker revealed the first untethered jailbreak for iOS 7.1 and showed a video of YouTube which showed the iPhone 4 running iOS 7.1 and it was jailbroken. It is thought that the jailbreak would work on other devices that have the same A4 chip.
The Apple iPhone 5 comes with the newer chip and people are waiting to see if the Evad3rs offer a jailbreak for it. Some people have said that they won’t and that they are waiting for the arrival of iOS 8 before they go ahead and jailbreak.
While we are waiting jailbreakers will have to make do with iOS 7 but they can make it more like iOS 7.1 thanks to tweaks courtesy of Cydia. iH8sn0w is also said to be working on a jailbreak for iOS 7.1 for the iPhone 4S.
For the time being if you want your iPhone to be jailbroken then you should stick with iOS 7 and avoid making he update.