You can jailbreak iOS 6.1.0 using the evasi0n v 1.1 which is the latest version to jailbreak 6.1.0. After release of evasi0n 6.1.0 jailbreak Apple has also seeded the iOS 6.1.1 and I believe that the existing untethered jailbreak shall be equally able to jailbreak iOS 6.1.1 untethered on iphone 5 and all other currently supported devices.
In evasi0n versio 1.0 there were mainly two bugs that caused a little bit panic for many new jailbreakers, the first bug was the “weather app not working after evasi0n jailbreak” and second was a longer boot time or freezing at the time of booting. These two issues were resolved in evasi0n 1.1 download. If you have already jailbroken your device your Cydia package will offer you to upgrade and if you want to jailbreak now you will get the latest download links for evasi0 1.1 from our evasi0n jailbreak download page.
Today it was reported on many tech blogs that evad3rs team already have sufficient exploits found in iOS 6.1 which they believe will remain in next versions i.e. 6.1.1 and they are confident that it will not be difficult to find out the iOS 6.1.1 Untethered Jailbreak once it is released, but still it is not recommended to update immediately after the release of 6.1.1 it is better to confirm whether a working 6.1.1 jailbreak is available or not. You can subscribe to our blog for updated news or keep visiting for latest jailbreak news etc.