iOS 10.1/10.1.1 Jailbreak is available with Luca's Yalu Jailbreak method. You can do Yalu Jailbreak online with zJailbreak. Otherwise, you can use Windows/mac computer to do this Jailbreak process with Cydia Impactor.
Yalu Online Jailbreak method is somewhat easier than Cydia Impactor method.
Also, Pangu8 Jailbreak app installer was compatible with iOS 10.1 to iOS 10.3.3 versions. Currently, Pangu8 jb app installer is disable. You can install Cydia and some more Jailbreak apps to your iOS 10.1 or 10.1.1 device using this installer.

Yalu Jailbreak for Full functional Cydia

Yalu102 jailbreak tool supports for the iOS 10.1 & iOS 10.1.1 Jailbreak. It is the only solution for these two versions to install full functional Cydia. You can use Yalu102 online method or Yalu102 PC required method for all devices except iPhone 7 and iPhone 7 Plus.
Yalu102 Online Guide
Yalu102 Jailbreak with PC
Extra_recipe+yaluX tool
Note - The same tool and the same Yalu 102 guide can be used for iOS 10 to iOS 10.2 Jailbreak. Still it has no untethered Pangu Jailbreak solution for iOS 10.1 and iOS 10.1.1 versions.

Yalu102 Online Guide

PC free online Yalu102 jailbreak solution is available to jailbreak iOS 10 to iOS 10.2 versions including iOS 10.1 & iOS 10.1.1 all the 64 bit device users. But iPhone 7 and iPhone 7 Plus running iOS 10.1.1 users can not use this method. They need to use Extra_recipe+yaluX method for Cydia.
Device Compatibility:
iPhone 7 & 7 Plus , iPhone 6S & 6S Plus ,iPhone 6 & 6 Plus

iPhone SE / iPhone 5s , iPod Touch 6G

iPad Mini 2 / iPad Mini 3 / iPad Mini 4

iPad Air /iPad Air 2 /iPad Pro
Step Guide
Step 01. Download Yalu Jailbreak app from this link first.
Please visit this page from your iOS device to install
   the Jailbreak for your iOS 10.1 or 10.1.1 device
Step 02. It will check the system compatibility.
Step 03. Click on the “zJailbreak install” button.
Step 04. It will ask device security code. Then click on the install button to have zJailbreak app on the device Home screen.
Step 05. Run the zJailbreak app.
Step 06. Choose “Yalu Full Jailbreak” from zJailbreak app store and Install it.
Step 07. Verify the zJailbreak app now. Go to Settings -- > General -- > Device Management-- > Open Yalu app -- > Verify -- > Trust.
Step 08. Go to Home screen and run the Yalu102 app. Then click on the Go button.
Step 09. Device will restart automatically to complete the jailbreak process having Cydia app on Home Screen.
Note -  If you get Cydia installation fail message, you have to repeat the above process several times.

Yalu102 Jailbreak with PC

Luca Todesco released Yalu102 tool version to jailbreak iOS 10 to iOS 10.2 including iOS 10.1 and iOS 10.1.1 versions. It supports all the 64 bit devices excluding iPhone 7 and iPhone 7 plus. It needs to get the PC support to complete this process.
Device compatibility:
iPhone 6s / iPhone 6s Plus / iPhone 6 / iPhone 6 Plus

iPhone 5s / iPhone SE

iPod Touch 6G
iPad Mini 2 / iPad Mini 3

iPad Air /iPad Air 2 /iPad Pro
Needs to get data backup using iCloud or iTunes.
Make sure to turn off Find my iPhone, Touch ID and the Passcode.
Download and install latest iTunes.
Download Yalu102 IPA from here.
Download Cydia Impactor according to your computer operating system from here.
Step Guide
Step 01 - Connect the device to the computer.
Step 02 - Open downloaded Cydia Impactor tool.
Step 03 - Drag & drop the downloaded Yalu102 IPA file into the Cydia Impactor.
Step 04 -  It will ask Apple ID and the Password.
Step 05 - It will be available Yalu102 app on your device home screen after complete the Apple ID sign in.
Step 06 -  Verify the Yalu102 app. Go to Settings --- > General --- > Device --- > Management --- > Trust.
Step 07 - Then run the Yalu102 app and click on the go button.
Finally, Cydia icon will be available on the device Home screen. If it is not, run the Yalu102 app and click on the go button again.

Extra_recipe+yaluX tool

Xerub & Ianbeer made this tool to jailbreak iPhone 7 running iOS 10.1.1 version only. This is a update tool version of yalu + mach_portal tool. You can use same Yalu102 jailbreak with PC method to complete this jailbreak. Click here to get the step guide. The only change is You have to use Extra_recipe+yaluX tool instead of Yalu102 IPA tool. Other steps remain as  same.
Download Extra_recipe+yaluX tool from here.

zJailbreak for latest non rooted app store apps.

zJailbreak is available for iOS 10.1 & iOS 10.1.1 versions. This tool provides non rooted App store and limited functional Cydia too.
zJailbreak for iOS 10.1.1 is very similar to iOS 10.2 zJailbreak guide. click here to get it.
Under the Z Jailbreak app, it has many useful third-party apps.  zJailbreak team will upgrade this more with adding many apps in future.  This method is the best solution to get paid app just for free.
Here are the Benefits of having Z Jailbreak apps on your iOS device.
AppStore paid apps can have for free.
Limited functional Cydia is available.
Easy to upgrade up to full functional Cydia, when having full jailbreak.
Cydia store apps can download.
Games hacked facility also available.
Watching & downloading movies for free.
Free online and offline music download.
FB skins can change.
Theme store brings brand new themes.
iOS 10.3.3
iOS 10.3.3 is the last version of iOS 10.. Apple released it without adding any new features. it includes bug fixes as well as major security update to prevent the iOS 10.3.3 jailbreak.
Online Jailbreak app Installer
Pangu8 has released online Jailbreak app installer for all latest iOS devices. It calls Pangu Anzhuang. You can use it to install some popular Jailbreak apps to iOS 10.3.3 and iOS 11-iOS 11.0.3 versions.
Yalu Jailbreak team is working to release Yalu 11 Jailbreak for the iOS 11 version including sub versions. Apple fixed Yalu Jailbreak releasing iOS 10.2.1.
Yalu Beta 7 IPA
Yalu beta 7 IPA is the Yalu's latest release. You can use this IPA to Jailbreak iOS 10.1 and 10.1.1 versions
Phoenix Jailbreak
Phoenix Jailbreak is one of the IPA based Jailbreak methods. It has been released to iOS 9.3.5
Yalu 102 - iOS 10.2 jailbreak
Luca Todesco released Yalu102 tool for iOS 10.2 jailbreak. This is the only solution to install full functional Cydia for iOS 10 to iOS 10.2 versions. Read more about iOS 10.2 jailbreak  and Yalu102 jailbreak.
iOS 10.1 bug bounty
Apple has introduced the bug bounty program to the iOS hackers. They offer millions of dollars to prevent the Jailbreak. Many Jailbreak hackers will be discouraged to release public Jailbreak because of this program

Jailbreak iOS 10.1/10.1.1

Extra_recipe+yaluX tool is recommended for iPhone 7 and iPhone 7 Plus running iOS 10.1.1 users only. However, these three methods provide semi tethered jailbreak solution. That is  lost Cydia every time, when reboot the device.Therefore, users must use Cydia reactive method to reactive Cydia after using given below methods.