Q: Why I can't take a screenshot in iOS 10.0.1 ?

You should be able to. It's working fine for me on iOS 10.0.1.

Restart your iPhone then try again.

1. Press and hold the Sleep/Wake button until the red slider appears.
2. Drag the slider to turn your device completely off
3. After the device turns off, press and hold the Sleep/Wake button again until you see the Apple logo.

Same for me.

Restart iPhone allows 1 screenshot.

A second screenshat can't me made until restart the iPhone again.

IPhone 5S und IPhone 6, both with 10.0.1

I have the same problem with iPhone 6 after upgrading to iOs 10.0.1 taking screen shots. I have switched off many times and switched on but the problem remains. The voice control appears, holding the Sleep/Wake button and pressing the start button instead of taking screen shot.

I'm having the same issue was hoping there was a solution to it.. I have the iPhone 6 too

I have the same issue on my iphone 5s. Cannot get the screen shot to work and it did before I updated to 10.0.1.

Did you try the procedure Carolyn outlined above?

Works fine for me.