Since I did a "clean" install of Ubuntu 16.04 LTS ("Xenial Xerus") on our three computers (all different models, from two different manufacturers), Steam simply will not install using the new (GNOME) Ubuntu Software.
When I go to the Steam website, download and run the Debian (".deb") installer, (GNOME) Ubuntu Software displays the blue "installing" bar, then this disappears and I'm left with an icon on the Launcher that says "Waiting to Install" when you hover over it.
There are no dialog boxes hiding in the background, and clicking the "Waiting to Install" icon does nothing...