Creating Recovery Media and Doing Image Backups in Windows 8.1/10

Warning: Many new PC's with Windows 8.1/10 pre-installed no longer ship with a printed Windows Product key card. The key is stored (hidden) in the PC bios chip. For more information see this article.

Windows 8.1 - Create a USB Recovery Drive
Microsoft Article on Creating a USB Stick to Recover Windows 8.1 (PC Boots OK)
(Note that Windows 8.1 does not allow creation of a CD/DVD recovery disc)
Microsoft Article on How to Create installation media for Windows 8.1 (PC Doesn't Boot)

Windows 10 - Create a USB Recovery Drive 
Microsoft Article on creating a recovery drive from Windows 10 (PC Boots OK)
Microsoft Article on Creating Windows 10 Installation Media (PC doesn't boot)

Windows 10 Education, Version 1703 - Microsoft Imagine
Microsoft Article on Creating a DVD or USB Stick to install Win 10 Ed Edition using Rufus
Download Rufus - The USB Utility for UEFI or BIOS PC's

Windows 8.1 - Create an Image Backup of Your Hard Disk*

Using DISM to Verify Component Store is Intact

Windows 10 - Create an Image Backup of Your Hard Disk*

1. Right click on Start Button, click "Control Panel" link
2. Click "file history" link
3. In lower left corner of File History window click "System Image Backup" link
4. In upper left of Backup and Restore window click "Create a System Image" link
5. Select the device to backup to and follow instructions in window...

* Before you start please read these important tips on doing image backups.

Also see this article from Microsoft on doing a backup with MS One Drive

Troubleshooting USB Flash Drive Problems

There can be lots of reasons why a USB flash drive fails or is no longer recognized by your PC.  So for starters, ALWAYS back up the data you store on a USB drive to cloud storage or your computer.  Having said that, the best article I have found that addresses most problems with USB drives (including a hardware failure of the drive itself) can be found here:

 Recover Files after USB Drive Corrupt or Unresponsive