I'm experiencing a serious problem with the new Ubuntu 18.04:
Dell XPS 13 with a freshly installed Ubuntu 18.04 LTS.
The touchpad provides only primary button (single click and double click), but no secondary button. In particular, when using the "test Your Settings" option in Settings -> Devices -> Mouse & Touchpad, any click on the touchpad is signaled as primary button, no matter where I click (left or right side of the touchpad). This remains the case even after I change the Primary Button from Left to Right (in Settings -> Devices -> Mouse & Touchpad -> General).
In comparison, an external mouse connected to my laptop (via usb) works fine: it provides both primary and secondary buttons with sides corresponding to the choice of Primary Button (in Settings -> Devices -> Mouse & Touchpad -> General).
Meanwhile, I found out that in addition to the solution proposed by EntropyWins, one can also control this touchpad behavior via the Tweaks tool (a.k.a. gnome-tweaks) under Keyboard & Mouse -> Touchpad -> Mouse Click Emulation. In any case, as EntropyWins explains in his answer, this is not a bug.