Can't change desktop environment

Solution 1:

I had the same problem. I solved it by adding a new user, then "Log Out" and connect again. It shows then the gear icon and you can change the environment.

Solution 2:

I had exactly this problem with Ubuntu 16.04. I solved it by installing a number of packages:

desktop-base
gnome-themes-standard
gnome-themes-standard-data
lightdm-gtk-greeter 
lightdm-gtk-greeter-settings
unity-greeter-badges

I also reinstalled: ubuntu-desktop and ubuntu-settings, but I don't think that was necessary. I now get a different login screen, but can choose the desktop environment from the top panel (drop down menu).