How do I allow non-root access to /ttyUSB0?

The device is most likely attached to user group dialout. Just add your user to the dialout group so you have appropriate permissions on the device.

sudo usermod -a -G dialout $USER

(You may need to logout and back in for the new group to take effect.)

No need to mess around with permissions or udev rules.

It's possible this is related to modemmanager, as detailed here.

I removed that (sudo apt-get remove modemmanager) and it fixed my problem.

You still need to be in the dialout group though.

sudo adduser <the user you want to add> dialout
sudo reboot

Mentioned by "Try now" worked for me. Check that You have dialout as group for ttyUSB0:

ls -l /dev/ttyUSB0

in my case the output is:

crw-rw---T 1 root dialout 188, 0 Feb 12 12:01 /dev/ttyUSB0